RESIDUAL ENTROPY, CONDITIONAL ENTROPY AND SUBSHIFT COVERS

MIKE BOYLE, DORIS FIEBIG, AND ULF FIEBIG

Abstract. We study the existence of subshift covers for topological dynamical systems, the infimum of the entropy jumps to such covers, and various aspects of conditional entropy and covering maps including a variational principle for covering maps. In particular we show every asymptotically h-expansive system (and therefore by Buzzi every C^{∞} homeomorphism of a compact Riemannian manifold) has a subshift cover of equal entropy. Our arguments in dimension zero are extended to higher dimension with theorems of Kulesza and Thomsen.

Contents

1. Introduction 1 2. Background and notation 3 3. Infinite residual entropy: an example 4 4. Conditional entropy of a homeomorphism 6 5. Conditional entropy of a quotient map 8 6. A variational principle for conditional entropy of a quotient map 9 7. Asymptotically h-expansive systems 14 8. Characterizing residual entropy in dimension zero 18 Appendix A. Zero dimensional covers 22 Appendix B. Zero dimensional covers of finite dimensional systems 25 Appendix C. Infinite residual entropy on a surface 29 Appendix D. Intermediate residual entropy 32 References 36

1. Introduction

Good subshift covers have been a useful tool for studying hyperbolic smooth dynamical systems (e.g. [Bow2]), and there is a nice theory of the abstract symbolic dynamics in this setting [F]. One would like to have some general understanding of which topological dynamical systems admit good symbolic dynamics.

The very first question, when is a system T the quotient of any subshift at all, turns out to be very difficult. An affirmative answer has long been known for expansive systems [Re] and for some or all group translations (e.g., Sturmian subshifts cover circle rotations), but we are aware of no general results on this question before the current paper and the work of Downarowicz [Do2]. (On the other hand, in Appendices A and B we see that the work of Kulesza and Thomsen gives excellent information about the existence of good quotient maps from zero dimensional systems onto given higher dimensional systems.) A necessary condition for T to admit a subshift cover is that T must have finite entropy, but this turns out to be not sufficient (Example 3.1 and [Do2]). We define the residual entropy ρ(T) of T as the infimum of the entropy gaps h(S) − h(T) over the set of subshifts S covering T. (If this set is empty, we set ρ(T) = ∞.)

Date: January 23, 2001.

2000 Mathematics Subject Classification. Primary: 37B10; Secondary: 37B40, 37C40, 37C45, 37C99, 37D35.

Key words and phrases. residual entropy, conditional entropy, entropy, variational principle, subshift, symbolic dynamics, cover, defect.

The research of the first author was supported by NSF Grant DMS9706852.

The residual entropy can be viewed as a descendant of the conditional topological entropy of a system, introduced by Misiurewicz [Mi2]. In Section 4 we review some essentials of the Misiurewicz development. In Section 5 we define the conditional topological entropy of a quotient map and work out some natural results. In Section 6 we prove a variational principle for the conditional entropy of a quotient map, describe its generalization by Downarowicz and Serafin, and give a counterexample to a natural simplifying conjecture. In Section 7, we characterize (Theorem 7.1) the existence of a quotient map from a mixing SFT S to a finite entropy product T of mixing shifts of finite type (SFTs). With this construction and the results on good zero dimensional extensions from the appendices, we go on to prove that any asymptotically h-expansive system T is a quotient of a subshift by a quotient map of conditional entropy zero (and in particular ρ(T) = 0). Buzzi [Bu], developing work of Yomdin [Y], has shown that any C^{∞} diffeomorphism of a compact Riemannian manifold is asymptotically h-expansive, and it follows that every such system has residual entropy zero.

The characterization of residual entropy turns out to be remarkably complicated. Here the best result, by far, is the Downarowicz characterization in the zero dimensional case [Do2], which we state in Section 8. The Downarowicz characterization is a mixed topological-measurable condition. In Section 8 we also characterize residual entropy in the zero dimensional case in terms of certain functions of words, without reference to measures. By analogy with the usual topological entropy, or even the conditional topological entropy of Misiurewicz, one expects that there should be reasonable definitions of residual entropy in terms of open sets or n, ϵ orbits; but we have been unable to achieve any such definition.

Among the open questions raised we single out two. First, if T has finite residual entropy, must there exist a subshift cover S such that ρ(T) = h(S)−h(T)? Second, to what extent is nonzero residual entropy compatible with smoothness? We know that a C^{∞} system has residual entropy zero, and in Appendix C we exhibit a finite entropy homeomorphism of a surface with infinite residual entropy. But for 1 ≤ k < ∞, we have no example of a C^{k} map with nonzero residual entropy, and we know of no obstruction to any value of residual entropy.

Some results of this paper (Example 3.1, Theorem 7.1, the infimum claim of Theorem 8.2, Proposition D.5, parts of Theorem 7.4 and A.1) were announced long ago [B2]. Downarowicz [Do1] came to the problem of residual entropy later but quite independently. In addition to giving the zero dimensional case characterization of residual entropy mentioned above, he gave examples of all allowable combinations of h(T), h^{∗}(T), ρ(T) [Do2]. The paper [Do2] also includes the characterization (done jointly with Boyle) of asymptotically h-expansive zero dimensional systems as subsystems of products of subshifts, and Downarowicz pointed out to us the utility of this characterization in simplifying our own proof that h^{∗}(T) = 0 implies ρ(T) = 0. The results common to this paper and [Do2] are proved with quite different methods.

We thank Downarowicz for his kind tolerance of our unpublished claims, for the proof simplification mentioned above, and for the stimulus to (finally) finish this work. The first named author also gratefully acknowledges support of the University of Washington in Seattle and the University of Heidelberg at different stages of this work.

2. Background and notation

Throughout the paper, by a system we will mean a selfhomeomorphism of a compact metrizable space, e.g. T: X → X. A subsystem of T is the restriction of T to a closed invariant subset of X. By the dimension of T we will mean the covering dimension of the domain X. For systems (X, T) and (Y, S), by a homomorphism φ: S → T we will mean a continuous map φ: X → Y such that φT = Sφ. An embedding φ: S ↪ T is an injective homomorphism; a quotient map φ: S ↠ T is a surjective homomorphism; an isomorphism or topological conjugacy is a bijective homomorphism.

The fixed point set of T will be denoted Fix(T) and the set of points of least period k will be denoted P_{k}^{o}(T). We let S ⟶^{per} T mean that for all positive integers n,

|Fix(S^{n})| > 0 ⟹ |Fix(T^{n})| > 0 .

(The condition S ⟶^{per} T is a necessary condition for S → T.) Similarly we let S ⟶^{iper} T mean that for all positive integers k,

|P_{k}^{o}(S)| ≤ |P_{k}^{o}(T)| .

(The condition S ⟶^{iper} T is a necessary condition for S ↪ T.)

Given a positive integer n, let A be a set of n elements (usually {0, …, n − 1}) with the discrete topology and let X = A^{ℤ} have the product topology. We view a point in X as a doubly infinite sequence x = … x_{−1}x_{0}x_{1} … with each x_{i} ∈ A. If T is the shift map on X, defined by requiring (T x)_{i} = x_{i+1}, then the system T is the full shift on n symbols. A subshift is a subsystem of some full shift. Any subshift may be described as the set of all points in some full shift in which a countable set of words does not occur. The subshift is a shift of finite type (SFT) if this set of excluded words can be chosen to be finite. For a thorough introduction to subshifts, see [LM, Ki, DGS].

A system T is expansive if there exists a metric d compatible with the topology such that there exists ϵ > 0 such that for each pair of points x, y with x ≠ y there exists some n in ℤ such that d(T^{n}x, T^{n}y) ≥ ϵ. If this condition holds for one compatible metric, then it holds for every compatible metric. Any zero dimensional expansive system is isomorphic to a subshift.

Now suppose that we have a sequence of systems T_{n}: X_{n} → X_{n} with bonding maps π_{n}: T_{n+1} → T_{n}. Let X = {x = (x_{1}, x_{2}, …) ∈ ∏_{n} X_{n}: π_{n}(x_{n+1}) = x_{n}, ∀n ∈ ℕ}. The inverse limit system T is the restriction to X of the infinite product T_{1} × T_{2} × . . .. It is elementary to check that every zero dimensional system is isomorphic to an inverse limit of subshifts. (This is an old fact, though for explicit proofs we only know the references [BH, T1], which also give additional structure).

Remark 2.1. The existence of an analogous inverse limit presentation is problematic in higher dimensions. First, a system need not be the inverse limit of expansive systems. For example, one easily checks that an expansive quotient of an isometry is finite, so no nontrivial (more than one point) isometry of a connected compact metric space is an inverse limit of expansive systems. Second, from the work of Elon Lindenstrauss [Li] we know that some infinite entropy systems are not inverse limits of finite entropy systems, and therefore are not inverse limits of subshifts or even quotients of subshifts.

In the sequel, to simplify notation, we will usually use the same symbol (e.g., T) to denote a selfhomeomorphism and its domain.

3. Infinite residual entropy: an example

The purpose of this section is to produce the following

Example 3.1. There is a selfhomeomorphism T of a compact metric space X such that h(T) < ∞ but ρ(T) = ∞.

(X, T) will be the inverse limit system formed from a sequence of mixing SFTs T_{n} and bonding maps π_{n}: T_{n+1} ↠ T_{n}. We will let p_{n} denote the projection X → X_{n}; so, π_{n}p_{n+1} = p_{n}, since for x = (x_{1}, x_{2}, …) we have π_{n}p_{n+1}x = π_{n}x_{n+1} = x_{n} = p_{n}x. We define the composition bonding maps π_{k,n}: T_{n} ↠ T_{k} by π_{k,n} = π_{k}π_{k−1} ⋯ π_{n−1}.

Choose the mixing SFTs and bonding maps to have the following properties (1) h(T_{n}) < h(T_{n+1}), n ≥ 1.

(2) sup_{n} h(T_{n}) < ∞ .

(3) There exists α > 1 such that for all k and for every finite orbit O of T_{k},

there exists n (depending on O) such that h(π_{k,n}^{−1}O) > logα. It remains to prove two claims.

CLAIM 1. A system with properties 1-3 exists.

CLAIM 2. (X, T) has finite entropy and infinite residual entropy.

Proof of CLAIM 2. Clearly h(T) = sup_{n} h(T_{n}) < ∞. To show ρ(T) = ∞, we argue by contradiction. Suppose S is a subshift and φ: S ↠ T. The map φ yields a commuting diagram of quotient maps in the following way. Define φ_{n}: S ↠ T_{n} as φ_{n} = p_{n}φ, then φ_{n} = π_{n}φ_{n+1} (since φ_{n} = p_{n}φ = π_{n}p_{n+1}φ = π_{n}φ_{n+1}) and more generally, φ_{k} = π_{k,n}φ_{n} whenever k < n. Now we have the key observation that for any n > k > 0 and finite T_{k}-orbit O,

h(φ_{k}^{−1}O) = h(φ_{n}^{−1}π_{k,n}^{−1}O) ≥ h(π_{k,n}^{−1}O) . (3.2)

Define

β^{∗} = sup{β ≥ 1 : h(φ_{k}^{−1}O) ≥ logβ , ∀k, ∀ finite T_{k} − orbits O} .

It follows immediately from (3.2) and property 3 that β^{∗} ≥ α > 1. Also β^{∗} < ∞ since logβ^{∗} ≤ h(S) < ∞.

Now fix any finite T_{k}-orbit O and any β such that 0 < β < β^{∗}. We will show that h(φ_{k}^{−1}O) ≥ log(αβ). (This implies β^{∗} ≥ αβ^{∗}, which gives the desired contradiction.) Using property 3, choose n such that h(π_{k,n}^{−1}O) > logα. The subshift E = π_{k,n}^{−1}O is SFT (because O and T_{n} are SFT), so h(E) is given by the growth rate of the periodic points of E, so for arbitrarily large N there are more than α^{N} orbits O′ of cardinality N. Fix such an N. For each such orbit O′ in E, h(φ_{n}^{−1}O′) ≥ logβ.

The map φ_{n} is a block code determined by some rule w_{i−M} … w_{i+M} ↦ (φ_{n}w)_{i}, where M depends on n but not w or i. For each orbit O′ in T_{n} of cardinality N, there is a set of T_{n}-words of length N, W(O′, T_{n}) = {x_{0} … x_{N−1} : x ∈ O′}. Let W(O′, S) = {w_{−M} … w_{N+M−1} : w ∈ φ_{n}^{−1}O′}. Because

h(φ_{n}^{−1}O′) ≤ \frac{log#W(O′, S)}{N + 2M} ,

we have #W(O′, S) ≥ β^{N+2M}. By the choice of M, for distinct O′ the sets W(O′, S) are disjoint. Therefore the number of S-words of length N + 2M in φ_{n}^{−1}E is at least

∑_{O′} β^{N+2M} ≥ α^{N}β^{N+2M} ,

where the sum is over the E-orbits O′ of cardinality N. Because N was arbitrarily large, we conclude

h(φ_{k}^{−1}O) = h(φ_{n}^{−1}E) ≥ lim_{N} \frac{log(α^{N}β^{N+2M})}{N + 2M} = log(αβ) .

□

Proof of CLAIM 1. There are many ways to find a sequence (T_{n}) satisfying (1) and (2), and in addition the condition that every T_{n} has a fixed point. For example, let T_{n} be a product S_{1} × S_{2} × ⋯ × S_{n}, where S_{n} is the mixing SFT whose defining matrix is the companion matrix of the polynomial x^{k+1} − x^{k} − 1, with k = n^{2}.

Now suppose we have such a sequence T_{n}. Fix α such that 0 < logα < lim_{n} h(T_{n}). Without loss of generality, suppose logα < h(T_{n}) for all n. The construction of the π_{n} is recursive. So suppose π_{1}, …, π_{n−1} are chosen; then we choose the map π_{n} as follows. For each k ≤ n and each orbit O of cardinality n or less in T_{k}, pick a finite orbit \overline{O} in T_{n} such that π_{k,n} sends \overline{O} to O. (For n = k, set \overline{O} = O.) Enumerate these orbits \overline{O} as \overline{O}_{1}, …, \overline{O}_{m}.

Let W be an SFT which is the disjoint union of irreducible SFTs W_{1}, …, W_{m} satisfying the following conditions (in which P_{k}^{o}(T) denotes the set of points in orbits of points of least period k in a subshift T):

• logα < h(W_{i}) < h(T_{n+1}), 1 ≤ i ≤ m,
• the period of \overline{O_{i}} divides the period of W_{i}, 1 ≤ i ≤ m,
• ∑_{i=1}^{m} #P_{k}^{o}(W_{i}) ≤ #P_{k}^{o}(T_{n}), k ∈ ℕ.

There are many ways to produce W. For example, irreducible SFTs W_{i} satisfying the entropy condition can be chosen, and then each W_{i} can be replaced by some W_{i} × P_{i}, where P_{i} is some finite orbit of sufficiently large cardinality which is divisible by #\overline{O_{i}}.

Now by Krieger’s Embedding Theorem [Kr2], we may identify W with a subsystem of T_{n}. Choose ψ_{i}: W_{i} ↠ \overline{O_{i}}, 1 ≤ i ≤ m, and let ψ: W ↠ ∪\overline{O_{i}} be the union of these maps. Because T_{n} is a mixing SFT with a fixed point, by the Extension Lemma (2.4 of [B1]) we may extend ψ to a quotient map π_{n}: T_{n+1} ↠ T_{n}.

The sequence (π_{n}) has the property 3. □

4. Conditional entropy of a homeomorphism

In this section, T is a selfhomeomorphism of a compact metric space. (Keep in mind our notational convention of using the same letter for a selfhomeomorphism and its domain.) First we recall the definition by open covers of the conditional topological entropy h^{∗}(T) of T, introduced by Misiurewicz [Mi2]. The basic idea of h^{∗}(T) is to give useful uniform estimates for conditional measure theoretic entropies. The definition is done in stages as follows.

(4.1)  N(U|B) = max_{V ∈B} min{card U′: U′ is a subcover of U|V }
(4.2)  h(T, U|B) = lim_{n} \frac{1}{n}logN(U_{0}^{n−1}|B_{0}^{n−1}) = inf_{n} \frac{1}{n}logN(U_{0}^{n−1}|B_{0}^{n−1})
(4.3)  h(T |B) = sup_{U} h(T, U|B) = lim_{U} h(T, U|B)
(4.4)  h^{∗}(T) = inf_{B} h(T |B) = lim_{B} h(T |B) .

Here, U and B represent open covers of T, and e.g. U_{0}^{n−1} denotes the open cover which is the common refinement of the covers T^{−i}U, 0 ≤ i ≤ n − 1. For a number a and a function α of open covers, the notation a = lim_{U} α(U) means that for any sequence U_{n} of open covers with mesh going to zero, lim_{n} α(U_{n}) = a. It is easy to see that

(4.5)  h^{∗}(T) ≤ h(T) ,
(4.6)  h^{∗}(T) = ∞ if and only if h(T) = ∞ , and
(4.7)  h^{∗}(R) ≤ h^{∗}(T) , for any subsystem R of T .

Next suppose T is zero dimensional; we will give a description of h^{∗}(T) using words in this case. Without loss of generality, suppose T is an inverse limit of subshifts T_{n} with surjective bonding maps π_{n}: T_{n+1} ↠ T_{n}. For n > k, let π_{k,n} denote the composition bonding map π_{k} ⋯ π_{n−1} : T_{n} ↠ T_{k}. Then for k < n we define

(4.8)  N(T_{n}, T_{k}, M) = max_{x∈T_{k}}{y_{0} … y_{M−1} : y ∈ T_{n}, π_{k,n}y = x}
(4.9)  h(T_{n}|T_{k}) = lim_{M} \frac{1}{M} log cardN (T_{n}, T_{k}, M)
(4.10)  h(T |T_{k}) = lim_{n} h(T_{n}|T_{k})

and then it is not difficult to verify

(4.11)  h^{∗}(T) = lim_{k} h(T |T_{k}) .

For context and meaning, we recall from [Bow1] the metric roots of conditional topological entropy. Recall, in a system T, a set C is an n, δ spanning set for K if for any x in K there exists y in C such that dist(T^{k}x, T^{k}y) ≤ δ for 0 ≤ k < n. For a compact (but not necessarily invariant) set K, the minimum cardinality of an n, δ spanning set for K is finite and is denoted by r_{n}(K, δ). The entropy of K is defined to be

(4.12) h(K) = lim_{δ→0} lim sup_{n→∞} (1/n)log r_{n}(K, δ) .

For n a nonnegative integer or n = ∞, and for ϵ > 0, define

B_{ϵ}^{n}(x) = {y: dist(T^{i}x, T^{i}y) ≤ ϵ, 0 ≤ i < n}

and let h^{∗}(x, ϵ) = h(B_{ϵ}^{∞}(x)). Bowen [Bow1] defined

h^{∗}(ϵ) = sup_{x∈T} h^{∗}(x, ϵ) .

In general the inequality lim_{ϵ→0} sup_{x∈T} h^{∗}(x, ϵ) ≥ sup_{x∈T} lim_{ϵ→0} h^{∗}(x, ϵ) can be strict, as in the following example.

Example 4.13. A system T in which sup_{x∈T} h^{∗}(x, ϵ) = log2 for all ϵ, and lim_{ϵ→0} h^{∗}(x, ϵ) = 0 for all x.

Description. Let S be the 2-shift and let O_{1}, O_{2}, … be an enumeration of the finite orbits of S. Let T_{0} = S, and for n ≥ 1 let T_{n} = S ∪ (S × ∪_{i=1}^{n}O_{i}). Thus T_{n} ⊂ T_{n+1} for all n. For n ≥ 1 define π_{n} : T_{n+1} → T_{n} by π_{n}(x) = x if x ∈ T_{n} and π_{n}(x) = z if x = (y,z) ∈ S ×O_{n+1}. It is not difficult to verify that the inverse limit system T constructed from the bonding maps π_{n} gives the required example. □

However, Bowen proved an inequality (Prop. 2.2 of [Bow1]) which implies the interchange of operations result

h^{∗}(ϵ) = lim_{δ→0} lim sup_{n}(1/n) max_{x∈T} log r_{n}(B_{ϵ}^{n}(x), δ) .

With this result, it is not very difficult (pp. 163-164 of [DGS] or Theorem 2.1 of [Mi2]) to verify the following claim: if U and B are open covers with ϵ > 0 such that every element of U has diameter less than ϵ and ϵ is a Lebesgue number for B (i.e. any ϵ-ball is contained in some element of B), then

h(T |U) ≤ h^{∗}(ϵ) ≤ h(T |B) .

It then follows easily that h^{∗}(T) = lim_{ϵ→0} h^{∗}(ϵ).

Bowen defined a system to be h-expansive (entropy expansive) if h^{∗}(ϵ) = 0 for some ϵ > 0. Bowen’s interest in [Bow1] was that this condition allowed computation of topological entropy from any open cover of sufficiently small mesh, and computation of measure theoretic entropy from any partition of sufficiently small mesh.

Misiurewicz [Mi1] defined a system to be asymptotically h-expansive in the case that lim_{ϵ→0} h^{∗}(ϵ) = 0. For such a system, Misiurewicz pointed out that μ ↦ h_{μ}(T) defines an uppersemicontinuous function on the compact space of T -invariant Borel probabilities, and in particular T has a measure of maximal entropy. Denker [De] finally characterized the finite entropy systems admitting a measure of maximal entropy by introducing as a further refinement of these ideas the local conditional topological entropy (see Ch. 20 of [DGS]).

We finish this section by recalling Ledrappier’s variational characterization of the conditional topological entropy of a selfhomeomorphism T of a compact metric space. (We will not apply this result, but it gives some context for Section 6.) Let T_{1} and T_{2} be two copies of T. For a T_{1} × T_{2} invariant Borel probability μ, let h(μ|T_{1}) denote the conditional measure theoretic entropy of T_{1} ×T_{2} with respect to the measure μ given the sigma algebra corresponding to projection onto T_{1}. Define

h^{∗}(m|T_{1}) = lim sup_{μ→m} h(μ|T_{1}) − h(m|T_{1}) ,  if h(m|T_{1}) is finite ,
= ∞ ,  if h(m|T_{1}) = ∞ .

Now we can state Ledrappier’s characterization from [Le]:

Theorem 4.14. [Le] Ledrappier Variational Principle:

h^{∗}(T) = max_{m} h^{∗}(m|T_{1}) .

5. Conditional entropy of a quotient map

In this section, S and T are selfhomeomorphisms of compact metric spaces and φ: S ↠ T. We assume the definitions and notation of the previous section. We define the conditional entropy of the quotient map φ to be

(5.1)  e^{∗}(φ) = inf_{V} h(S|φ^{−1}V) = lim_{V} h(S|φ^{−1}V)

where V represents an arbitrary open cover of T. If φ = Id_{S}, then e^{∗}(φ) = h^{∗}(S).

In the case that S and T are zero dimensional inverse limit systems of sequences of subshifts S_{n} and T_{n}, we can give a description of e^{∗}(φ) with words as follows. We let p_{n} denote the projection T ↠ T_{n} or S ↠ S_{n}, and similarly let π_{n} denote a given bonding map S_{n+1} ↠ S_{n} or T_{n+1} ↠ T_{n}. For y ∈ T_{k}, let N(S_{n}, T_{k}, M, y) be the cardinality of {(p_{n}x)_{0} … (p_{n}x)_{M−1} : x ∈ S and p_{k}φx = y}. Set

(5.2)  N(S_{n}, T_{k}, M) = max_{y∈T_{k}} N(S_{n}, T_{k}, M, y)
(5.3)  h(S_{n}|T_{k}) = lim_{M} \frac{1}{M} log N(S_{n}, T_{k}, M)
(5.4)  h(S|T_{k}) = lim_{n} h(S_{n}|T_{k})

and then it is not difficult to verify

(5.5)  e^{∗}(φ) = lim_{k} h(S|T_{k}) .

Above, if S is a subshift then we may regard each S_{n} as S, and the limit over n is unnecessary.

The following properties are evidence for the reasonableness of the definition of the conditional entropy of a quotient map.

Facts 5.6. Suppose S and T are selfhomeomorphisms of compact metric spaces, and φ: S ↠ T. Then the following hold.

(1) e^{∗}(φ|R) ≤ e^{∗}(φ), for any subsystem R of S.

(2) h(φ^{−1}R) ≤ h(R) + e^{∗}(φ), for any subsystem R of T.
(3) h_{μ}(S) ≤ h_{φ_{∗}μ}(T) + e^{∗}(φ), for any S-invariant Borel probability μ.
(4) max{e^{∗}(φ), e^{∗}(ψ)} ≤ e^{∗}(φψ) ≤ e^{∗}(φ) + e^{∗}(ψ), for any quotient map ψ.
(5) h^{∗}(S) ≤ e^{∗}(φ).
(6) h^{∗}(T) ≤ h^{∗}(S) + e^{∗}(φ).
(7) h^{∗}(T) ≤ 2e^{∗}(φ).

Proof. We will verify the sixth property and leave the other verifications to the reader. Let U be an open cover of S and let V, B be open covers of T. Then

(5.7)  N(V_{0}^{n−1}|B_{0}^{n−1}) = N((φ^{−1}V)_{0}^{n−1}|(φ^{−1}B)_{0}^{n−1})

≤ N((φ^{−1}V)_{0}^{n−1}|U_{0}^{n−1}) · N(U_{0}^{n−1}|(φ^{−1}B)_{0}^{n−1}) (5.8) and so

(5.9)  h(T, V|B) ≤ h(S, (φ^{−1}V)|U) + h(S, U|(φ^{−1}B))

≤ h(S|U) + h(S|(φ^{−1}B)) . (5.10)

Therefore

(5.11)  h^{∗}(T) = inf_{B} sup_{V} h(T, V|B)

≤ h(S|U) + e^{∗}(φ) (5.12)

and because U was arbitrary we have

(5.13)  h^{∗}(T) ≤ inf_{U} h(S|U) + e^{∗}(φ) = h^{∗}(S) + e^{∗}(φ) .
□

Remark 5.14. One easily sees that when e^{∗}(φ) = 0, all the inequalities in (5.6) above become equality. In particular, if e^{∗}(φ) = 0, then both S and T must be asymptotically h-expansive.

6. A variational principle for conditional entropy of a quotient map

In this section we will establish a variational principle for the conditional entropy of a quotient map; briefly describe the Downarowicz-Serafin and Ledrappier-Walters conditional variational principles; and give Example 6.11, a quotient map with positive entropy jumps on measures but not subsystems.

For a quotient map φ: S ↠ T, we use the notation

SM(φ) = sup_{m} (h_{m}(S) − h_{φ_{∗}m}(T))

where the supremum is taken over the S-invariant Borel probabilities. Given m and finite m-measurable partitions P, Q we let H_{m}(P |Q) = H_{m}(P ∨ Q) − H_{m}(Q), the measure theoretic conditional entropy of P given Q. We first observe that the well known concavity of the map m ↦ H_{m}(P) implies the concavity of m ↦ H_{m}(P |Q).

Lemma 6.1. ([DS]; Lemma 3.2 of [LeW]) Suppose P and Q are finite measurable partitions, μ and ν are probabilities, 0 < λ < 1, and m = λμ + (1 − λ)ν. Then

λH_{μ}(P |Q) + (1 − λ)H_{ν}(P |Q) ≤ H_{m}(P |Q) .

Proof. We can assume the sets B in Q have positive μ and ν measure. Define

λ_{B} = λμ(B) / [λμ(B) + (1 − λ)ν(B)]

and let e.g. μ_{B} denote the conditional measure, μ_{B}(C) = μ(B ∩ C)/μ(B). Then

λH_{μ}(P |Q) + (1 − λ)H_{ν}(P |Q) = ∑_{B∈Q} λμ(B)H_{μ_{B}}(P) + (1 − λ)ν(B)H_{ν_{B}}(P)

= ∑_{B}[λμ(B) + (1 − λ)ν(B)][λ_{B}H_{μ_{B}}(P) + (1 − λ_{B})H_{ν_{B}}(P)]
≤ ∑_{B}[λμ(B) + (1 − λ)ν(B)][H_{λ_{B}μ_{B}+(1−λ_{B})ν_{B}}(P)] = H_{m}(P |Q) .

□ Lemma 6.2. Suppose S and T are finite entropy selfhomeomorphisms of zero dimensional compact metric spaces, φ: S ↠ T is a quotient map, and e^{∗}(φ) is the topological conditional entropy of the quotient map. Then

SM(φ) ≥ e^{∗}(φ) − h^{∗}(T) .

Proof. Let ϵ > 0. We show that there is an S-invariant Borel probability measure m such that

h_{m}(S) − h_{φ_{∗}m}(T) > e^{∗}(φ) − h^{∗}(T) − 2ϵ .

For a clopen partition α of S let α_{n} denote α ∨ S^{−1}α ⋯ ∨ S^{−(n−1)}α. Similarly, for a clopen partition β of T set β_{n} := β ∨ T^{−1}β ⋯ ∨ T^{−(n−1)}β. Then

h_{m}(S, α) − h_{φ_{∗}m}(T, β) = lim_{n→∞} \frac{1}{n}H_{m}(α_{n}|φ^{−1}β_{n})

if α is finer than φ^{−1}β.

Now choose a clopen partition β of T, a clopen partition α of S which is finer than φ^{−1}β, and an integer N such that for all n ≥ N it holds that

|e^{∗}(φ) − \frac{1}{n}logN(α_{n}|φ^{−1}β_{n})| < ϵ and
|h^{∗}(T) − h(T |β)| < ϵ.

For each n ≥ N, fix a set B_{n} ∈ β_{n} such that

#{A ∈ α_{n}|A ⊂ φ^{−1}B_{n}} = N(α_{n}|φ^{−1}β_{n}) .

Let E_{n} ⊂ S such that for each A ∈ α_{n} with A ⊂ φ^{−1}B_{n}, it holds that #(E_{n} ∩ A) = 1. Let

σ_{n} = \frac{1}{#E_{n}} ∑_{x∈E_{n}} δ_{x} ,

where δ_{x} denotes the point mass at x. Observe that σ_{n}(φ^{−1}B_{n}) = 1. Then

H_{σ_{n}}(α_{n}|φ^{−1}β_{n}) = − ∑_{B∈β_{n}} σ_{n}(φ^{−1}B) · ∑_{A∈α_{n}} σ_{n}(A|φ^{−1}B)logσ_{n}(A|φ^{−1}B)

= −σ_{n}(φ^{−1}B_{n}) · ∑_{x∈E_{n}} \frac{1}{#E_{n}} · log\frac{1}{#E_{n}}

= log#E_{n} .

Fix an integer q. For n > q large enough, for 0 ≤ b < q and a ≥ 1, write n = aq + b. Then for 0 ≤ j < q it holds that

α_{n} = α_{j} ∨ S^{−j}α_{q} ∨ S^{−(q+j)}α_{q} ∨ ⋯ ∨ S^{−((a−2)q+j)}α_{q} ∨ S^{−((a−1)q+j)}α_{b+q−j}.

Thus for each 0 ≤ j < q we have for Q(j) = {0 ≤ k < j} ∪ {(a − 1)q + j ≤ k < n} and σ_{n,k} := (S^{k})_{∗}σ_{n} that

log#E_{n} ≤ H_{σ_{n}}(α_{n}|φ^{−1}β_{n})

≤ ∑_{r=0}^{a−2} H_{σ_{n}}(S^{−(rq+j)}α_{q}|φ^{−1}T^{−(rq+j)}β_{q}) + ∑_{k∈Q(j)} H_{σ_{n}}(S^{−k}α|φ^{−1}T^{−k}β)

= ∑_{r=0}^{a−2} H_{σ_{n,rq+j}}(α_{q}|φ^{−1}β_{q}) + ∑_{k∈Q(j)} H_{σ_{n,k}}(α|φ^{−1}β)

≤ ∑_{r=0}^{a−2} H_{σ_{n,rq+j}}(α_{q}|φ^{−1}β_{q}) + 3q · log(#α)

Adding these q inequalities, dividing by n and appealing to Lemma 6.1 we get

q · \frac{1}{n}log#E_{n} ≤ \frac{1}{n} ∑_{p=0}^{n−1} H_{σ_{n,p}}(α_{q}|φ^{−1}β_{q}) + 3q^{2} · \frac{1}{n}log(#α)

≤ H_{μ_{n}}(α_{q}|φ^{−1}β_{q}) + 3q^{2} · \frac{1}{n}log(#α)

where μ_{n} = 1/n ∑_{k=0}^{n−1}(S^{k})_{∗}σ_{n}. Thus

\frac{1}{n}log#E_{n} < \frac{1}{q}H_{μ_{n}}(α_{q}|φ^{−1}β_{q}) + 3q · \frac{1}{n}log(#α) .

For a suitable subsequence n_{i} and a measure m we have that μ_{n_{i}} → m and, since the sets of the finite partitions α_{q} and φ^{−1}β_{q} are closed open, we get thus that

lim_{n→∞} \frac{1}{n}log#E_{n} ≤ \frac{1}{q}H_{m}(α_{q}|φ^{−1}β_{q}) .

Since this holds for all q, we conclude that

lim_{n→∞} \frac{1}{n}log#E_{n} ≤ h_{m}(S, α) − h_{φ_{∗}m}(T, β) .

We thus have e^{∗}(φ) − ϵ ≤ h_{m}(S, α) − h_{φ_{∗}m}(T, β), by the choice of the partitions α and β. Therefore

e^{∗}(φ) − ϵ ≤ h_{m}(S) − h_{φ_{∗}m}(T, β) . (6.3)

Now if β′ is a clopen partition of T finer than β, then

|h_{φ_{∗}m}(T, β) − h_{φ_{∗}m}(T, β′)| = lim_{n→∞} \frac{1}{n}H_{φ_{∗}m}(β′_{n}|β_{n})

≤ lim_{n→∞} \frac{1}{n}logN(β′_{n}|β_{n})

= h(T, β′|β) .

Thus h_{φ_{∗}m}(T, β) ≥ h_{φ_{∗}m}(T, β′) − h(T, β′|β) for every partition β′ finer than β. Therefore

h_{φ_{∗}m}(T, β) ≥ h_{φ_{∗}m}(T) − h(T |β) ≥ h_{φ_{∗}m}(T) − (h^{∗}(T) + ϵ) , so
h_{φ_{∗}m}(T, β) ≥ h_{φ_{∗}m}(T) − h^{∗}(T) − ϵ .

Using this last inequality to substitute for h_{φ_{∗}m}(T, β) in (6.3), we get

h_{m}(S) − h_{φ_{∗}m}(T) ≥ e^{∗}(φ) − h^{∗}(T) − 2ϵ

as required. □

Lemma 6.4. Suppose φ_{1} : S_{1} ↠ T, φ_{2} : S_{2} ↠ T, and F is the fibered product of S_{1} and S_{2} by φ_{1} and φ_{2}, with projections p_{1} : F ↠ S_{1} and p_{2} : F ↠ S_{2}. Then SM(φ_{1}) = SM(p_{2}) and SM(φ_{2}) = SM(p_{1}).

Remark 6.5. We only appeal to this lemma in the case SM(φ_{1}) = 0, which follows from the easier inequality SM(φ_{1}) ≥ SM(p_{2}).

Proof of Lemma 6.4. Suppose μ, μ_{1}, μ_{2}, \overline{μ} are Borel probabilities on F, S_{1}, S_{2}, T with φ_{1}μ_{1} = \overline{μ} = φ_{2}μ_{2}, p_{1}μ = μ_{1}, and p_{2}μ = μ_{2}. Let A, B_{1}, B_{2}, C be the Borel σ-algebras of F, S_{1}, S_{2}, T. Then

h_{μ_{1}}(S_{1}, B_{1}) − h\overline{μ}(T, C) = h_{μ_{1}}(S_{1}, B_{1}|φ_{1}^{−1}C) = h_{μ}(F, p_{1}^{−1}B_{1}|p_{1}^{−1}φ_{1}^{−1}C)

= h_{μ}(F, p_{1}^{−1}B_{1}|p_{2}^{−1}φ_{2}^{−1}C) ≥ h_{μ}(F, p_{1}^{−1}B_{1}|p_{2}^{−1}B_{2}) (6.6)

= h_{μ}(F, p_{1}^{−1}B_{1} ∨ p_{2}^{−1}B_{2}|p_{2}^{−1}B_{2}) = h_{μ}(F) − h_{μ_{2}}(S_{2}) .

It follows that SM(φ_{1}) ≥ SM(p_{2}). For the other direction, given φ_{1} : μ_{1} ↦ \overline{μ}, choose μ_{2} such that φ_{2}μ_{2} = μ, and choose μ to be the relatively independent joining of μ_{1} and μ_{2} [Ru]. Then the inequality in (6.6) becomes equality, and it follows that SM(φ_{1}) = SM(p_{2}). Likewise of course, SM(φ_{2}) = SM(p_{1}). □

Recall the definition of Ledrappier [Le]: φ: S ↠ T is a principal extension of T if h_{μ}(S) = h_{φμ}(T) for every S-invariant Borel probability μ. A system S has a principal extension to a zero dimensional system if S is finite dimensional (by Theorem B.2) or if S is asymptotically h-expansive (by Corollary A.2 and Facts 5.6). We expect that all finite entropy selfhomeomorphisms of compact metric spaces admit principal extensions to zero dimensional systems.

Theorem 6.7. Variational Principle for Quotient Maps. Suppose φ: S ↠ T, where T is asymptotically h-expansive, h(S) < ∞ and S admits a zero dimensional principal extension. Then

e^{∗}(φ) = SM(φ) .

Proof. We have e^{∗}(φ) ≥ SM(φ) (without the hypotheses on S and T) by Facts (5.6). It remains to prove the reversed inequality.

By Corollary A.2, there is a zero dimensional extension π_{1} : T_{1} ↠ T such that h^{∗}(T_{1}) = e^{∗}(π_{1}) = 0. The map π_{1} is a principal extension by Facts 5.6. Let π_{2} : S_{2} ↠ S be the assumed zero dimensional principal extension of S. Let F be the fibered product of T_{1} and S_{2} by the maps π_{1} and φπ_{2}, with projections p_{1} : F ↠ T_{1} and p_{2} : F ↠ S_{2}. By the last lemma, SM(p_{2}) = SM(π_{1}) = 0 and SM(p_{1}) = SM(φπ_{2}). Because SM(π_{2}) = 0, we have SM(φπ_{2}) = SM(φ), so SM(p_{1}) = SM(φ). We also have

e^{∗}(p_{1}) ≥ e^{∗}(φ) − e^{∗}(π_{1})

since e^{∗}(π_{1}) + e^{∗}(p_{1}) ≥ e^{∗}(π_{1}p_{1}) = e^{∗}(φπ_{2}p_{2}) ≥ e^{∗}(φ). Finally, because F and T_{1} are zero dimensional, by appeal to Lemma 6.2 we get

SM(φ) = SM(p_{1}) ≥ e^{∗}(p_{1}) − h^{∗}(T_{1})
≥ e^{∗}(φ) − e^{∗}(π_{1}) − h^{∗}(T_{1})
= e^{∗}(φ) .
□

Remark 6.8. We can have h^{∗}(T) > 0 with e^{∗}(φ) at either end of the interval [SM(φ), SM(φ) + h^{∗}(T)]. For example, if φ is the identity map, then SM(φ) = 0 and e^{∗}(φ) = h^{∗}(T); whereas if φ is projection of S = R × T onto T, then SM(φ) = h(R) = e^{∗}(φ).

Remark 6.9 (Downarowicz-Serafin Relative Variational Principle). Downarowicz and Serafin have proved a more general variational principle, in a manuscript [DS] we received after finishing the writing above of Theorem 6.7, near the completion of this paper. For a quotient map φ: S → T, they defined the topological conditional entropy of (X,S) given the factor (Y,T), h(X|Y), which in our notation is given by

h(X|Y) = sup_{U} inf_{B} h(S, U|φ^{−1}B)

where B ranges over open covers of Y and U ranges over open covers of X. This contrasts with the definition

e^{∗}(φ) = inf_{B} sup_{U} h(S, U|φ^{−1}B).

As explained in [DS], it is not difficult to check that the relation of the two definitions is given by

h(X|Y) ≤ e^{∗}(φ) ≤ h(X|Y) + h^{∗}(T) ,

so e^{∗}(φ) = h(X|Y) in the case that h^{∗}(T) = 0, and therefore our Theorem 6.7 can be obtained as a corollary of their result.

Remark 6.10 (Ledrappier-Walters Relative Variational Principle). Ledrappier and Walters [LeW] proved a relative variational principle for pressure for a quotient map φ: S ↠ T, which for entropy takes the form

sup_{μ} h_{μ}(S) = h_{ν}(T) + ∫ h(S, π^{−1}(y))dν(y)

where the supremum is taken over all invariant measures μ such that π_{∗}μ = ν. Subsequently Walters extended these developments and others in [W1].

We finish this section with an example, which in particular shows that one cannot simplify the proof of Lemma 6.2 by using a drop of topological entropy of a suitable restricted map.

Example 6.11. There are transitive subshifts S and T and a quotient map f: S → T with the following properties:

(1) For every subsystem R of T (including R = T), h(f^{−1}R) = h(R).

(2) e^{∗}(f) > 0.

We first define T as a subshift of {1, 2, 3, 4, 5}^{ℤ}, which is obtained from the following skeleton construction. Let x denote a symbol not in {1, 2, 3, 4, 5}. Skeletons will be blocks with symbols in {1, x}. The skeleton of order 0 is s_{0} := 1. The skeleton of order 1 is s_{1} := (s_{0}x)^{4}s_{0} = s_{0}xs_{0}xs_{0}xs_{0}xs_{0}, in which the 0-skeleton occurs 4^{1} + 1 times. Inductively, the k + 1-skeleton is s_{k+1} = (s_{k}x^{k+1})^{4^{k+1}}s_{k} = s_{k}x^{k+1}s_{k}x^{k+1} … x^{k+1}s_{k}, in which the k-skeleton occurs 4^{k+1} + 1 times. Now we replace the symbols x in the skeletons by some symbol from {2, 3, 4, 5} as follows. In the first step, replace each occurrence of the symbol x in the 1-skeleton s_{1} by symbols from {2, 3, 4, 5} such that every symbol from {2, 3, 4, 5} occurs, and call the resulting block s_{1}^{1}. This block has symbols in {1, 2, 3, 4, 5}; for example, s_{1} could be the block s_{1}^{1} := 121314151. In the second step, first replace in s_{2} every occurence of s_{1} with s_{1}^{1}, to obtain (s_{1}^{1}x^{2})^{4^{2}}s_{1}^{1}, and then replace in this block each occurence of x^{2} with an element from {2, 3, 4, 5}^{2} such that each of these 2-blocks is used. Call the resulting block s_{2}^{1}; it has symbols in {1, 2, 3, 4, 5}. Inductively, for k ≥ 2, first replace in s_{k+1} each of the 4^{k+1} + 1 occurences of the block s_{k} with s_{k}^{1} to obtain the block (s_{k}^{1}x^{k+1})^{4^{k+1}}s_{k}^{1}, and then replace the blocks x^{k+1} with elements from {2, 3, 4, 5}^{k+1} such that each of the elements from {2, 3, 4, 5}^{k+1} is used. Call the resulting block s_{k+1}^{1}. In this way we obtain a family of blocks (s_{k}^{1})_{k≥1} with symbols in {1, 2, 3, 4, 5}. Now suppose t ∈ {1, 2, 3, 4, 5}^{ℤ}. Then by definition t ∈ T if and only if for every n ≥ 0 there is some k ≥ 1 such that t[−n, n] is a subblock of s_{k}^{1}.

Define a 1-block map g: {0, 1, 2, 3, 4, 5}^{ℤ} → {1, 2, 3, 4, 5}^{ℤ} by g(y)_{0} = 1 if y_{0} ≤ 1 and g(y)_{0} = y_{0} if y_{0} ≥ 2. Let S := g^{−1}(T) and let f: S → T be the restriction of g to S.

Since {2, 3, 4, 5}^{ℤ} is contained in T, we have h(T) ≥ log4. To get an upper estimate for the entropy of S, consider the subshift T′ with symbols {1, x} such that a point t is in T′ if every subblock t_{i} … t_{j} is contained in some skeleton s_{k}. Consider w = w_{1} … w_{n} ∈ B_{n}(T′) which sees at least two 1’s and let m(w) = max{p|∃i, w_{i} … w_{i+p+1} = 1x^{p}1}. Thus w is a subblock of x^{m(w)+1}s_{m(w)}x^{m(w)+1}. Therefore the first occurrence of 1x^{m(w)}1 in w and the first and last occurence of the symbol 1 in w determine the whole block w. Thus there are at most n^{3} blocks w in B_{n}(T′) such that m(w) = m for a fixed m > 0. There are at most n + 1 blocks in B_{n}(T′) which do not see at least two 1’s. Since m(w) < n, it follows that #B_{n}(T′) ≤ n + 1 + n(n^{3}) ≤ 3n^{4}.

For 0 ≤ k ≤ n, let B_{n,k}(T) denote the set of T -blocks of length n in which the symbol 1 occurs exactly k times. Then

#B_{n}(S) = ∑_{k=0}^{n} #B_{n,k}(T) · 2^{k} ≤ ∑_{k=0}^{n} #B_{n}(T′) · 4^{n−k} · 2^{k}

≤ ∑_{k=0}^{n} 3n^{4} · 4^{n−k} · 2^{k} ≤ 3n^{5} · 4^{n}.

Thus log4 ≥ h(S) ≥ h(T) ≥ log4.

Now let R be a subshift of T. First consider the case that there is a t ∈ R with t_{0} = 1. Let n_{1} > |s_{1}^{1}|. Since t[−n_{1}, n_{1}] is a subblock of some s_{k}^{1} and t_{0} = 1, we get that t[−n_{1}, n_{1}] contains s_{1}^{1}. Consider n_{2} > n_{1} + |s_{2}^{1}|. Then t[−n_{2}, n_{2}] is a subblock of some s_{k}^{1}, and since t[−n_{1}, n_{1}] contains s_{1}^{1}, we get that t[−n_{2}, n_{2}] contains s_{2}^{1}. Inductively we see in this way that every s_{k}^{1} is a subblock of t. Thus t has a dense orbit in T and thus R = T and f^{−1}R = S. If t ∈ R implies that t_{i} ≠ 1 for all i, then R is contained in {2, 3, 4, 5}^{ℤ} and thus every point in R has a unique preimage. In any case, h(f^{−1}R) = h(R).

Simple estimates show that the relative frequency of the symbol 1 in every kskeleton block is greater than 1/4, thus we get e^{∗}(f) ≥ (1/4)log2 > 0. This completes the example.

7. Asymptotically h-expansive systems

In this section we will show that an asymptotically h-expansive system has residual entropy zero. The heart of the argument is the coding construction used in the next result.

Theorem 7.1. Suppose S is a mixing SFT and T_{1}, T_{2}, … is a sequence of mixing SFT’s such that h(T_{n}) > 0 for all n. Let T be the product system T_{1} × T_{2} × . . .. Then the following are equivalent.

(1) There exists a quotient map φ: S ↠ T.

(2) h(T) < h(S) and S ⟶^{per} T.

Proof. (1) ⟹ (2) Clearly h(T) ≤ h(S) and S ⟶^{per} T. To rule out the possibility h(S) = h(T), we will appeal to the following result (Corollary 6.8 in [BT]): for a given mixing SFT S, the set of entropies of its uniform mixing SFT quotients is finite. (Here V is a uniform quotient of S if there exists a quotient map ψ: S ↠ V such that ψ_{∗}: max_{S} ↦ max_{V}, where e.g. max_{S} denotes the unique measure of maximal entropy of S.)

So, suppose h(T) = h(S) and φ: S ↠ T. Each T_{n} has a unique measure of maximal entropy μ_{n}, and because the product system T has finite entropy it follows that μ = ∏_{n} μ_{n} is the unique measure of maximal entropy of T. There exists an S-invariant Borel probability ν such that φ_{∗}: ν ↦ μ (Prop. 3.11 of [DGS]), and this measure ν must satisfy h_{ν}(S) ≥ h_{μ}(T). Because h_{ν}(S) ≤ h(S) = h(T) = h_{μ}(T), the only possibility is that ν = max_{S}. For each n, by postcomposing φ with projection onto T_{n} we see that T_{n} is a uniform quotient of S. But the set {h(T_{n}) : n ∈ ℕ} is infinite, since the numbers h(T_{n}) are positive and sum to h(T) < ∞, and this is a contradiction.

(2) ⟹ (1) Let R_{0} denote S. For n ≥ 1, choose mixing SFTs R_{n} to satisfy the following conditions:

(i) ∑_{k=n+1}^{∞} h(T_{k}) < h(R_{n}) < h(R_{n−1}) − h(T_{n})
(ii) T_{n} × R_{n} ⟶^{iper} R_{n−1}
(iii) lim_{n} h(R_{n}) = 0 .

(The choice of R_{n} may be carried out recursively as follows. Given h(R_{n−1}) > ∑_{k=n}^{∞} h(T_{k}), we have h(R_{n−1})−h(T_{n}) > ∑_{k=n+1}^{∞} h(T_{k}), so we can choose a mixing SFT satisfying (i), and for (iii) also satisfying h(R_{n}) − ∑_{k=n+1}^{∞} h(T_{k}) < 1/n. Now by (i), h(T_{n} × R_{n}) < h(R_{n−1}), so |O_{k}(T_{n} × R_{n})| < |O_{k}(R_{n−1})| except for perhaps finitely many k. The inequality can be achieved for all k by replacing R_{n} with a suitable equal entropy mixing SFT cover, by appeal to the Covering Lemma 2.1 of [B1].)

For n ≥ 1, R_{n} is a mixing SFT; h(T_{n} × R_{n}) < h(R_{n−1}); and T_{n} × R_{n} ⟶^{iper} R_{n−1}. Therefore, by Krieger’s Embedding Theorem [Kr2], we may choose for each n ≥ 1 an embedding i_{n} from T_{n} × R_{n} into R_{n−1}. Then define embeddings j_{n} : T_{1} × ⋯ × T_{n} × R_{n} ↪ S by composition:

j_{1} : (t_{1}, r_{1}) ↦ i_{1}(t_{1}, r_{1})
j_{2} : (t_{1}, t_{2}, r_{2}) ↦ i_{1}(t_{1}, i_{2}(t_{2}, r_{2}))
j_{3} : (t_{1}, t_{2}, t_{3}, r_{3}) ↦ i_{1}(t_{1}, i_{2}(t_{2}, i_{3}(t_{3}, r_{3})))
j_{4} : (t_{1}, t_{2}, t_{3}, t_{4}, r_{4}) ↦ i_{1}(t_{1}, i_{2}(t_{2}, i_{3}(t_{3}, i_{4}(t_{4}, r_{4}))))

and so on. Regarding j_{n} as an isomorphism to a subsystem S_{n} of S, for n ≥ k let p_{k,n} denote the map S_{n} ↠ T_{k} defined by following j_{n}^{−1} with the projection π_{k} onto T_{k}. For every n ≥ k and x ∈ S_{n}, we have p_{k,n}(x) = π_{k}(i_{1}i_{2} ⋯ i_{k})^{−1}(x), so for n > k it holds that p_{k,n} equals the restriction of p_{k,n−1} to S_{n}. Also for every n ≥ k, the map p_{1,n} × ⋯ × p_{k,n}: S_{n} → T_{1} × ⋯ × T_{k} is surjective.

For each n ≥ 1, extend the map p_{n,n}: S_{n} ↠ T_{n} to some quotient map φ_{n}: S ↠ T_{n}. This is possible by the Extension Lemma 2.4 of [B1] because T_{n} is a mixing SFT and S ⟶^{per} T_{n}. If n ≥ k, then the restriction to S_{n} of the map φ_{1} × ⋯ × φ_{k} agrees with the surjection p_{1,1} × ⋯ × p_{k,k}: S_{n} ↠ T_{1} × ⋯ × T_{k}. Let φ = ∏_{n=1}^{∞} φ_{n}. Then it follows from compactness that the map φ: S → T is surjective as required. □

Remark 7.2. Let us note that some obvious candidate conditions are not sufficient to ensure that T is a quotient of a shift of finite type. Suppose T is an inverse limit T_{1} ↞ T_{2} ↞ T_{3} … of mixing SFT’s T_{n}, with each bonding map T_{n+1} ↠ T_{n} finite to one and noninjective. It is not difficult to verify that h^{∗}(T) = 0, and therefore, by Theorem 7.4, T is a quotient of a subshift of equal entropy. However, regardless of whether T has a fixed point, T is not the quotient of any SFT (Theorem 2.10 of [B3]).

The following result is the key ingredient in the proof of Theorem 7.4.

Lemma 7.3. Suppose T_{1}, T_{2}, … are subshifts, T = T_{1} × T_{2} × ... , and h(T) < ∞. Then there is a subshift V and a quotient map ψ: V ↠ T such that e^{∗}(ψ) = 0. In particular, h(V) = h(T) and ρ(T) = 0.

Proof. Let T′_{n} be a mixing SFT with a fixed point such that T_{n} is isomorphic to a subsystem of T′_{n} and h(T′_{n}) < h(T_{n}) + 2^{−n}. Then T is isomorphic to a subsystem of T′ = ∏ T′_{n} and h(T′) < ∞. It follows from Facts 5.6 that the collection of systems for which the conclusion of the theorem holds is closed under passage to subsystems. So without loss of generality, we may assume each T_{n} is a mixing SFT, and there is a mixing SFT S such that h(S) > h(T) and S ⟶^{per} T.

Now we return to the end of the proof of Theorem 7.1 and continue from there. For n ≥ 1, φ maps the subshift S_{n} of S onto T, and the S_{n} are a nested sequence S_{1} ⊃ S_{2} ⊃ · · ·. Let V = ∩S_{n}. It follows from compactness that φ maps the subshift V onto T. Let ψ be the restriction of φ to V. (Remark: already we know ρ(T) = 0, because h(V) = lim h(S_{n}) = h(T).)

Fix k in ℕ. Let U = U_{k} = T_{1} × ⋯ × T_{k}; so for u ∈ U and i ∈ ℤ, we have u_{i} = (t_{i}^{(1)}, …, t_{i}^{(k)}). By (5.5),

e^{∗}(ψ) = lim_{k} lim_{M} \frac{1}{M} log N(V, U_{k}, M) .

Let j = j_{k}. Now S_{k} ⊃ V and we have

U × R_{k} ↠^{j} S_{k} ↠ U
(u, r) ↦ s ↦ u

in which the map S_{k} ↠ U is φ followed by the projection p_{k} onto U. Because j is a surjective block code, there is a positive integer J such that for any s, s′ in S_{k} with s_{0} … s_{n−1} ≠ s′_{0} … s′_{n−1}, there exist (u, r), (u′, r′) in U × R_{k} and i in [−J, n − 1 + J] such that j(u, r) = s and j(u′, r′) = s′ and (u_{i}, r_{i}) ≠ (u′_{i}, r′_{i}). Therefore, for every u in U,

#{v_{0} … v_{M−1} : (p_{k}ψv)_{i} = u_{i}, 0 ≤ i ≤ M − 1}
≤ #{(u_{−J}, r_{−J}) … (u_{M−1+J}, r_{M−1+J}): (p_{k}φj(u, r))_{i} = u_{i}, 0 ≤ i ≤ M − 1}
≤ #{r_{−J} … r_{M−1+J} : r ∈ R_{k}} · #{(u_{−J} … u_{−1})(u_{M} … u_{M−1+J}): u ∈ U}

where the last inequality follows from p_{k}φj: (u, r) ↦ u. Consequently

lim_{M} \frac{1}{M} log N(S, U_{k}, M) ≤ h(R_{k})

and e^{∗}(ψ) ≤ lim_{k} h(R_{k}) = 0. □

The following theorem was proved independently in the zero dimensional case by Downarowicz [Do2]. (The information on e^{∗}(φ) is not in his statement but can be derived from his construction.)

Theorem 7.4. Suppose T is an asymptotically h-expansive selfhomeomorphism of a compact metric space. Then there exists a subshift S and a quotient map φ: S ↠ T with e^{∗}(φ) = 0. In particular, h(S) = h(T) and ρ(T) = 0.

Proof. By Corollary A.2, there is an asymptotically h-expansive zero dimensional system T′ and a quotient map β: T′ ↠ T such that e^{∗}(β) = 0. It was shown in [Do2] that any zero dimensional asymptotically h-expansive system embeds in a finite entropy product of subshifts, so without loss of generality we may assume that T′ is a subsystem of a system T′′ such that h(T′′) < ∞ and T′′ is a product of subshifts. By Lemma 7.3, there is a subshift S′′ and a quotient map α: S′′ ↠ T′′ such that e^{∗}(α) = 0. Define S = α^{−1}(T′), define α′ as the restriction of α to S, and define φ = βα′. Using Facts 5.6, we have

e^{∗}(φ) = e^{∗}(βα′) ≤ e^{∗}(β) + e^{∗}(α′) ≤ e^{∗}(β) + e^{∗}(α) = 0 ,

so e^{∗}(φ) = 0 and h(S) = h(T). □

Remark 7.5. We thank Downarowicz, who pointed out to us [Do1] that when T is zero dimensional asymptotically h-expansive, ρ(T) = 0 follows easily from the product-of-subshifts case, by appeal to the characterization of asymptotically h-expansive zero dimensional systems as subsystems of products of subshifts [Do2]. Our original more complicated proof still appealed to this special case but also used additional marker arguments. What is really needed for those additional arguments is cleanly isolated by the product-of-subshifts characterization in [Do2].

Corollary 7.6. If h(T) = 0, then there is a zero entropy subshift S and a quotient map φ: S ↠ T.

Remark 7.7. Theorem 7.4 shows that h^{∗}(T) = 0 implies ρ(T) = 0. But there is no general inequality between h^{∗}(T) and ρ(T). In Example 3.1 we have T such that 0 < h^{∗}(T) < ∞ = ρ(T). On the other hand it is not difficult to construct from a mixing SFT S an inverse limit T of mixing SFTs such that S ↠ T and h(S) = h(T) (so ρ(T) = 0) but h^{∗}(T) > 0.

Buzzi [Bu], extending work of Yomdin [Y], showed that if T is a C^{r} selfmap of a compact Riemannian m-dimensional manifold with boundary, then h^{∗}(T) ≤ \frac{m}{r}logR(T), where R(T) is the spectral radius of the map DT on the tangent bundle. (Actually, Buzzi defined a local entropy h_{loc}(T), and this appears in his formula where we use h^{∗}(T). We avoid this h_{loc}(T) because it is always equal to h^{∗}(T), and because there has been some conflicting usage of the term “local entropy”: Newhouse [Ne] used a different (possibly equivalent?) definition, while Brin and Katok ([BrKa], [Ma2]) earlier gave the term a different meaning.)

Buzzi’s result implies that a C^{∞} system is asymptotically h-expansive. So we have the following immediate corollary to Theorem 7.4 and the theorem of Buzzi.

Theorem 7.8. A C^{∞} diffeomorphism of a compact Riemannian manifold has residual entropy zero.

In Appendix C we give an example of a homeomorphism of a disc which has finite entropy and infinite residual entropy. For 1 ≤ r < ∞, we have no results on the compatibility of positive residual entropy with C^{r} smoothness.

The considerations above show that some reasonable symbolic dynamics exist for a C^{∞} system. A much harder problem is to construct them in an explicit and useful way.

8. Characterizing residual entropy in dimension zero

We begin with the Downarowicz characterization. Let T be a zero dimensional system, presented as an inverse limit of subshifts T_{n} by surjective bonding maps. Let M(T) denote the compact convex set of T -invariant Borel probabilities. Each μ in M(T) projects to a measure μ_{n} in M(T_{n}). Define functions h_{n} on M(T) by

h_{n}(μ) = h_{μ_{1}}(T_{1})  if n = 1 ,
= h_{μ_{n}}(T_{n}) − h_{μ_{n−1}}(T_{n−1})  if n > 1 ,

so h_{μ}(T) = ∑ h_{n}(μ). Let F denote the set of sequences of continuous functions f_{n}: M(T) → ℝ such that f_{n} ≥ h_{n}, and let ||f|| denote the supremum of |f|. Now we can state the Downarowicz characterization.

Theorem 8.1. [Do2] Let the notation be as above for a zero dimensional system T. Then

ρ(T) = inf_{F} || ∑_{n=1}^{∞} f_{n}|| − h(T) .

The rest of this section is devoted to a different (and much more modest) characterization of the residual entropy of T, also just for the case that T is zero dimensional, and also in terms of some given presentation of T as an inverse limit of subshifts T_{n} with surjective bonding maps π_{n}: T_{n+1} ↠ T_{n}. Without loss of generality, assume that the alphabets of the T_{k} are disjoint and assume that the maps π_{n} are one-block codes (if x and y are in T_{n+1} and x_{0} = y_{0}, then (π_{n}x)_{0} = (π_{n}y)_{0}). Let W_{k}(T_{n}) denote the set of words of length k occurring in T_{n}, let W(T_{n}) = ∪_{k}W_{k}(T_{n}), let W(T) = ∪_{n}W(T_{n}). By a word oracle for T we will mean a function α: W(T) → ℕ = {1, 2, … } satisfying the following two properties (in which α_{n} denotes the restriction of α to W(T_{n})):

• (Submultiplicative Property) If the concatenation W_{1}W_{2} is in W(T_{n}), then α_{n}(W_{1}W_{2}) ≤ α_{n}(W_{1}) · α_{n}(W_{2})

• (Extension Property) There exist positive constants c_{1}, c_{2}, … such that for all n and all W in W(T_{n}),

∑ α_{n+1}(W′) ≤ c_{n}α_{n}(W)

where the sum is over all the words W′ such that π_{n}(W′) = W.

Given a word oracle α, we define

h(α_{n}) = lim_{k} \frac{1}{k} log ∑_{W ∈W_{k}(T_{n})} α_{n}(W)

(where the finite limit exists as a consequence of the submultiplicative property). For any k, n, as a consequence of the extension property we have

c_{n} ∑_{W ∈W_{k}(T_{n})} α_{n}(W) ≥ ∑_{W ∈W_{k}(T_{n+1})} α_{n+1}(W)

and therefore h(α_{n}) ≥ h(α_{n+1}) for all n. We define the entropy of the word oracle α to be h(α) = lim_{n} h(α_{n}).

Theorem 8.2. Let T be a zero dimensional system, with notations as above. Then the set of entropies of subshift covers of T equals the set of entropies of word oracles for T. In particular, if T has finite entropy then ρ(T) = inf_{α} h(α) − h(T), where the infimum is over all word oracles for T.

Proof. First, given a subshift cover φ: S ↠ T , we will define a word oracle α for T such that h(α) = h(S).

For any quotient map ψ: R ↠ V of subshifts and W ∈ W_{j}(V), we define

ψ^{−1}W = {x_{0} … x_{j−1} : x ∈ R, (ψx)_{0} … (ψx)_{j−1} = W } .

Let φ_{n} = p_{n}φ (i.e., φ_{n} is φ followed by projection onto T_{n}). Define α = ∪α_{n} by setting α_{n}(W) = |φ_{n}^{−1}W |. Clearly the Submultiplicative Property holds for α.

For the Extension Property, let r = r_{n+1} be a coding radius for φ_{n+1} (if j is any nonnegative integer, then x_{−r} … x_{j+r−1} determines (φ_{n+1}x)_{0} … (φ_{n+1}x)_{j−1}). Set c_{n} = |W_{r}(S)|^{2}. Fix any j and any word W in W_{j}(T_{n}). For any U in φ_{n}^{−1}W, there are at most c_{n} words W′ in π_{n}^{−1}W such that U ∈ φ_{n+1}^{−1}W′. Also,

φ_{n}^{−1}W = ⋃{φ_{n+1}^{−1}W′: W′ ∈ π_{n}^{−1}W } .

Therefore

∑_{W′∈π_{n}^{−1}W} α_{n+1}(W′) = ∑_{W′∈π_{n}^{−1}W} |φ_{n+1}^{−1}W′|

≤ c_{n} | ⋃_{W′∈π_{n}^{−1}W} φ_{n+1}^{−1}W′| = c_{n}|φ_{n}^{−1}W | = c_{n}α_{n}(W) .

Therefore the extension condition holds for α. It is not difficult to check that h(α_{n}) = h(S) for all n, giving h(α) = h(S).

For the remaining, more difficult inclusion, let a word oracle α be given. We will construct for T a subshift cover whose entropy equals h(α). We will use notation of the following sort: x[i, i + j) denotes the word x_{i}x_{i+1} … x_{i+j−1} of length j.

Let S be a mixing SFT with entropy h(S) satisfying h(α_{1}) < h(S). Also suppose S is a 1-step SFT, i.e., if x_{0} … x_{i} and y_{i} … y_{j} are S-words with x_{i} = y_{i}, then x_{0} … x_{i}y_{i+1} … y_{j} is an S-word. Let Z be a zero entropy subshift containing no periodic points. We will pick certain nested subshifts

(Z × S)_{1} ⊃ (Z × S)_{2} ⊃ (Z × S)_{3} ⊃ ⋯

of Z × S, and for each n define a quotient map ψ_{n} from (Z × S)_{n} onto some subshift containing T_{n}, such that ψ_{1} × ⋯ × ψ_{n} maps (Z × S)_{n} onto a supersystem of (p_{1}×⋯×p_{n})(T). Let (Z ×S)_{∞} = ∩_{n}(Z ×S)_{n}. Then it is clear from compactness that ∏_{n=1}^{∞} ψ_{n} = ψ maps (Z × S)_{∞} onto a supersystem of T. We will arrange that h(Z × S)_{∞} = h(α) and that ψ^{−1}(T) is a subsystem of full entropy in (Z × S)_{∞}.

For this scheme, we will choose in Z certain nested clopen (marker) sets F_{1} ⊃ F_{2} ⊃ · · · . For each n, there will be (large) positive integers N_{n} and P_{n} with N_{n} < P_{n} such that with (F, N, P) = (F_{n}, N_{n}, P_{n}) the following marker conditions are satisfied:

(8.3) the clopen sets Z^{i}F are disjoint for 0 ≤ i < N, and

∪_{i=0}^{P −1} Z^{i}F = Z . (8.4)

(In the last line, on the left Z is the map and on the right Z is the space.) First we define F_{1}. Choose δ > 0 such that h(α_{1}) + log(1 + δ) < h(S). Pick some symbol a from W_{1}(S). Let W_{j}^{a}(S) be the set of S-words U of length j such that U begins with a and Ua is an S-word. Using the fact that S is a mixing SFT, we pick N_{1} in ℕ such that (using the notation ⌈x⌉ = min{k ∈ ℤ : k ≥ x}) we have the following:

(8.5)  |W_{j}^{a}(S)| ≥ ⌈(1 + δ)^{j}⌉ ∑_{W ∈W_{j}(T_{1})} α_{1}(W) ,  j ≥ N_{1} .

(The condition (8.5) will be used to guarantee ψ_{1} has image containing T_{1}.) Then set P_{1} = 2N_{1} and by the standard argument (see [Kr2] or [B1]), we choose a set F_{1} in Z satisfying the marker conditions (8.3)-(8.4) with (F, N, P) = (F_{1}, N_{1}, P_{1}).

Next, we give the recursive definition for (F_{n+1}, N_{n+1}, P_{n+1}), supposing that (F_{n}, N_{n}, P_{n}) has been defined. First we choose N_{n+1} > P_{n} such that

(8.6)  (1 + \frac{δ}{n})^{j} > c_{n}⌈(1 + \frac{δ}{n + 1})^{j}⌉ ,  j ≥ N_{n+1} .

Then we set N′ = N_{n+1} + P_{n} and choose a set F′ in Z such that (F, N, P) = (F′, N′, 2N′) satisfies the marker conditions (8.3)-(8.4). Finally (to achieve F_{n+1} ⊂ F_{n}), for z ∈ F′ we let h(z) = min{i ≥ 0: Z^{i}z ∈ F_{n}}, and define F_{n+1} = {Z^{h(z)}z: z ∈ F′}. Let P_{n+1} = 2N′ + P_{n}. Then F_{n+1} ⊂ F_{n} and (F, N, P) = (F_{n+1}, N_{n+1}, P_{n+1}) satisfies the marker conditions (8.3)-(8.4). This finishes the definition of the marking sets F_{n}.

When Z^{i}x ∈ F_{n} and Z^{i+j}x ∈ F_{n} and Z^{k}x ∉ F_{n} for i < k < i+j, then we say the integer interval [i, i + j) is an F_{n}-marker block (of length j) for x. An F_{n}-marker block [i, i + j) is a normalized F_{n}-marker block if i = 0. For each n, there are only finitely many normalized F_{n}-marker blocks. If [i, i + j) is an F_{n}-marker block for x, then we say [0, j) is a normalized F_{n}-marker block for x at i. A point x in Z via F_{n} produces a tiling of the integers by normalized F_{n}-marker blocks.

For each n, let R_{n} denote the full shift on the symbols of T_{n}, and let π_{n}: R_{n+1} → R_{n} denote the one-block code given by the one-block coding rule which defines π_{n}: T_{n+1} → T_{n}.

For a normalized F_{1}-marker block B = [0, j), fix a subset W_{B}^{a,1}(S) of W_{j}^{a}(S) such

that #W_{B}^{a,1}(S) = ⌈(1 + δ)^{j}⌉ · ∑_{W} α_{1}(W). Define a map ψ_{B}^{a,1} : W_{B}^{a,1}(S) → W_{j}(T_{1})

such that #(ψ_{B}^{a,1})^{−1}(W) = ⌈(1+δ)^{j}⌉·α_{1}(W) for all W ∈ W_{j}(T_{1}). This is possible by (8.5). Then define (Z × S)_{1} to be the set of points (x, y) in Z × S satisfying the following condition: if B = [0, j) is a normalized F_{1}-marker block of x at i, then y[i, i + j) ∈ W_{B}^{a,1}(S). Finally, set ψ_{1}(x, y)[i, i + j) = ψ_{B}^{a,1}(y[i, i + j)), when B = [0, j) and B is a normalized F_{1}-marker block of x at i. This defines the map ψ_{1} : (Z × S)_{1} → R_{1}. Obviously T_{1} ⊂ ψ_{1}(Z × S)_{1}.

The recursive step (the definition of ψ_{n+1} assuming the definitions of ψ_{1}, …, ψ_{n}) is the main step of the proof and for this we must endure some further notation for the marker structure. For x ∈ Z, if [i, i + j) is an F_{n}-marker block, then x determines a factorization of [i, i + j) as a concatenation of F_{n−1}-marker blocks (if n > 1), a concatenation of F_{n−1}-marker blocks into F_{n−2}-marker blocks (if n > 2), and so on. We call this whole structure an F_{[1,...,n]}-marker block (of length j). Formally, if [i, i + j) is an F_{n}-marker block of x, then the F_{[1,...,n]}-marker block B of x at i is the n-tuple B = (B_{1}, …, B_{n}), where B_{n} = [i, i + j) and for 1 ≤ k < n, B_{k} is the set of F_{k}-marker blocks of x which are contained in [i, i + j). If B is an F_{[1,...,n]}-marker block of x at i, then we define its normalization B′ to be the F_{[1,...,n]}-marker block of Z^{i}x at 0, and we say B′ is the normalized F_{[1,...,n]}-marker block of x at i. For each n, there are only finitely many normalized F_{[1,...,n]}-marker blocks.

Next we state the inductive hypothesis for the recursive argument. We suppose, for each 1 ≤ k ≤ n and for each normalized F_{[1,...,k]}-marker block B of length j, that a subset W_{B}^{a,k}(S) ⊂ W_{j}^{a}(S) and a map ψ_{B}^{a,k}: W_{B}^{a,k}(S) → W_{j}(T_{k}) are given such that the following properties hold:

(1) For each 1 ≤ k ≤ n and each normalized F_{[1,...,k]}-marker block B of length j it holds that #(ψ_{B}^{a,k})^{−1}(W′) = ⌈(1 + δ/k)^{j}⌉ · α_{k}(W′) for each W′ ∈ W_{j}(T_{k}).

(2) For each 1 ≤ k < n and each normalized F_{[1,...,k+1]}-marker block B it holds that W_{B}^{a,k+1}(S) ⊂ W_{b(1)}^{a,k}(S) ∗ ⋯ ∗ W_{b(l)}^{a,k}(S), where b(1) ⋯ b(l) is the factorization of B into F_{[1,...,k]}-marker blocks.

(3) For each k with 1 ≤ k < n, and each normalized F_{[1,...,k+1]}-marker block B with factorization b(1) ⋯ b(l) into F_{[1,...,k]}-marker blocks, it holds for each U = U_{1} ⋯ U_{l} ∈ W_{B}^{a,k+1}(S) with U_{i} ∈ W_{b(i)}^{a,k}(S) that π_{k}(ψ_{B}^{a,k+1}(U)) =

ψ_{b(1)}^{a,k}(U_{1}) ∗ ⋯ ∗ ψ_{b(l)}^{a,k}(U_{l}).

We shall now define for each normalized F_{[1,...,n+1]}-marker block B of length j a subset W_{B}^{a,n+1}(S) ⊂ W_{j}^{a}(S) and a map ψ_{B}^{a,n+1} : W_{B}^{a,n+1}(S) → W_{j}(T_{n+1}) such that the inductive hypothesis holds with n+1 in place of n. Let B = (B_{1}, …, B_{n+1}) be a normalized F_{[1,...,n+1]}-marker block, where B_{n+1} = [0, j) and b(1) ⋯ b(l) is the factorization of B into F_{[1,...,n]}-marker blocks of lengths j_{1}, …, j_{l}. For each W ∈ W_{j}(T_{n}) it holds that

#{U = U_{1} ⋯ U_{l} ∈ W_{b(1)}^{a,n}(S) ∗ ⋯ ∗ W_{b(l)}^{a,n}(S): ψ_{b(1)}^{a,n}(U_{1}) ∗ ⋯ ∗ ψ_{b(l)}^{a,n}(U_{l}) = W }

≥ ⌈(1 + \frac{δ}{n})^{j}⌉ · α_{n}(ψ_{b(1)}^{a,n}(U_{1})) · · · · · α_{n}(ψ_{b(l)}^{a,n}(U_{l}))

≥ ⌈(1 + \frac{δ}{n})^{j}⌉ · α_{n}(W)

≥ ⌈(1 + \frac{δ}{n + 1})^{j}⌉ · ∑_{W′} α_{n+1}(W′) ,

where the first inequality holds by the induction hypothesis (1); the second holds by the submultiplicitivity of α_{n}; and, because j > N_{n+1}, the last holds by (8.6) and the extension property of α_{n}.

Thus we can choose a set W_{B}^{a,n+1}(S) ⊂ W_{b(1)}^{a,n}(S) ∗ ⋯ ∗ W_{b(l)}^{a,n}(S) such that

#W_{B}^{a,n+1}(S) = ⌈(1+δ/(n+1))^{j}⌉·∑_{W′} α_{n+1}(W′) and a map ψ_{B}^{a,n+1} : W_{B}^{a,n+1}(S) →

W_{j}(T_{n+1}) such that #(ψ_{B}^{a,n+1})^{−1}(W′) = ⌈(1 + δ/(n + 1))^{j}⌉ · α_{n+1}(W′) for each W′ ∈ W_{j}(T_{n+1}) and such that π_{n}(ψ_{B}^{a,n+1}(U) = ψ_{b(1)}^{a,n}(U_{1}) ∗ ⋯ ∗ ψ_{b(l)}^{a,n}(U_{l}) for all U = U_{1} … U_{l} ∈ W_{B}^{a,n+1}(S). This new family of subsets of S-blocks and maps satisfies the induction hypotheses with n+1 in place of n. This finishes the recursive step.

For n ≥ 1 we define the subshift (Z × S)_{n+1} := {(x, y) ∈ Z × S: if B is a normalized F_{[1,...,n+1]}-marker block of length j of x at i, then y[i, j) ∈ W_{B}^{a,n+1}(S)}. Define ψ_{n+1}(x, y)[i, j) = ψ_{B}^{a,n+1}(y[i, j)) if B is a normalized F_{[1,...,n+1]}-marker block of length j of x at i. This defines a map ψ_{n+1} : (Z×S)_{n+1} → R_{n+1}. Obviously T_{n+1} ⊂ ψ_{n+1}(Z ×S)_{n+1} and π_{n}ψ_{n+1}(x, y) = ψ_{n}(x, y) for all (x, y) ∈ (Z ×S)_{n+1}.

Next, we check that (Z × S)_{∞} has entropy equal to h(α). Suppose x′ ∈ Z and x′ has an F_{[1,...,n]}-marker block of length j at i and W ∈ W_{j}(T_{n}). Then

|{(x′, y)[i, i + j): (x′, y) ∈ (Z × S)_{n} and (ψ_{n}(x′, y))[i, i + j) = W }|

= ⌈(1 + \frac{δ}{n})^{j}⌉α_{n}(W) .

Here N_{n} ≤ j ≤ P_{n} and therefore when n > 1 we have

0 ≤ \frac{1}{j}log⌈(1 + \frac{δ}{n})^{j}⌉ ≤ \frac{1}{j}log((1 + \frac{δ}{n − 1})^{j}) = log(1 + \frac{δ}{n − 1}) := γ(n)

(where the second inequality holds because the c_{n} in (8.6) are positive integers). Because h(Z) = 0, after considering concatenations of W ’s we conclude

h(α_{n}) ≤ h((Z × S)_{n}) ≤ h(α_{n}) + γ(n) .

Because lim_{n} γ(n) = 0, we conclude

h(α) = lim h(α_{n}) = lim h((Z × S)_{n}) = h((Z × S)_{∞}) .

It only remains to see that the subsystem ψ^{−1}(T) of (Z × S)_{∞} has full entropy. Suppose that (x, y) is a point in (Z × S)_{∞} such that (ψ_{1} × ψ_{2} × …)(x, y) ∉ T. Since π_{n}ψ_{n+1}(x, y) = ψ_{n}(x, y) for all n, there is thus n_{0} such that ψ_{n}(x, y) ∉ T_{n} for all n ≥ n_{0}. Thus there is k ≥ 0 such that ψ_{n_{0}}(x,y)[−k,k] ∉ W (T_{n_{0}}). Thus, since the π_{n} are 1-block maps, ψ_{n}(x, y)[−k, k] ∉ W (T_{n}) for all n ≥ n_{0}. Therefore, for all n ≥ n_{0}, the interval [−k, k] is not contained in an F_{n}-marker block of x. Thus, there is some i in [−k, k] such that Z^{i}x ∈ F_{n} for all n ≥ n_{0}. This shows that (Z × S)_{∞} − ψ^{−1}(T) is contained in the set E = ∪_{i} ∪_{N≥1} ∩_{n≥N}(Z × S)^{−i}(F_{n} × S). Because the sets (Z × S)^{j}(F_{n} × S) are disjoint for 0 ≤ j < N_{n}, we have that for every invariant measure μ and every n, μ(F_{n}×S) ≤ 1/(N_{n}). Because lim_{n} N_{n} = ∞, we conclude that E has measure zero with respect to any invariant probability, and it follows then from the variational principle that h(ψ^{−1}(T)) = h((Z × S)_{∞}.

This concludes the proof.

□

Appendix A. Zero dimensional covers

The next result, without the condition on e^{∗}(φ), was proved independently by the first author and Klaus Thomsen by essentially the same construction. (Without any entropy condition, this is an old result of Anderson [A].) Thomsen’s result was circulated in the preprint [T1] (and he considers more generally systems T which are continuous but not necessarily injective or surjective). In [B2] the first author announced his result, which eventually appeared as the supporting result Prop. 2.5 in [GW]. We revisit the construction below because we want to establish the inequality for e^{∗}(φ); the basic construction is unchanged, but additional argument and care are required.

Theorem A.1. Suppose T is a selfhomeomorphism of a compact metric space Y and h(T) < ∞. Then there is a selfhomeomorphism S of a zero dimensional compact metric space X and a quotient map φ: S ↠ T such that h(S) = h(T) and moreover h^{∗}(S) ≤ e^{∗}(φ) ≤ h^{∗}(T).

Proof. PART I. In this part we describe an ingredient of the construction.

Let P be a finite open cover of Y. Given a positive integer N, let C be a minimum cardinality subcover of the common refinement ⋁_{i=0}^{N−1} T^{−i}P. Let C = {C_{1}, …, C_{m}}. For each C_{i}, fix a choice of elements P (i, j) of P such that C_{i} = ∩_{j=0}^{N−1}T^{−j}P (i, j). For 1 ≤ i ≤ m, let W (i) be the word C_{i}00 … 0, where C_{i} is followed by N − 1 zeros (and so W (i) has length N.) Let S′ be the subshift on all concatenations of the words W (i) (so, S′ is conjugate to the tower of height N over the m-shift). Note, for large N the entropy of S′ will be close to the entropy of T with respect to the open cover P.

To each x in S′, associate a bisequence \overline{x} as follows. If x_{t} ⋯ x_{t+N−1} = C_{i}00 ⋯ 0, then set \overline{x}_{t+j} = P (i, j). Define K(x) = ∩_{n∈ℤ}\overline{T^{−n}\overline{x}_{n}}, a closed (possibly empty) subset of T. Now define S(P, C, N) as the subshift on {x ∈ S′: K(x) ≠ ∅}. This subshift has entropy at most h(S′); also, T is covered by the sets K(x), x ∈ S(P, C, N).

PART II. For a suitable refining sequence of open covers P_{n} of Y, we will construct as above S_{n} := S(P_{n}, C_{n}, N_{n}). For each n > 0, we will define bonding maps S_{n} → S_{n−1}. This will give us an inverse limit system S. For a point x in S, x = (x^{(1)}, x^{(2)}, …) , the sets K(x^{(n)}) will be nested and their intersection will be a point, y. The desired quotient map φ: S ↠ T will be defined by sending x to y.

The construction is recursive. For n = 0, let P_{0} be the cover {X} of X and let N_{0} = 1; so, S_{0} is the one-point subshift. Notation: given an open cover U = {U_{1}, …, U_{m}}, we let U_{i}^{∗} be the union of the sets U_{j} such that U_{i} ∩ U_{j} ≠ ∅, and we let U^{∗} denote the open cover {U_{1}^{∗}, …, U_{m}^{∗}}.

Fix ϵ_{1} > ϵ_{2} > …, some arbitrary sequence of positive numbers decreasing to zero, and suppose the construction has been carried out for 0, …, n − 1. Choose a finite open cover P := P_{n} such that

• P refines C_{n−1} (every element of P is contained in some element of C_{n−1}),

• the mesh of P is less than ϵ_{n},

• h(T) − h(T, P) < ϵ_{n}, and

• h(T |P) − h^{∗}(T) < ϵ_{n} . Then choose N = N_{n} and C = C_{n} such that

• N is a multiple of N_{n−1}, and

• C is a minimum cardinality subcover of the join of P, …, T^{−(N−1)}P such that |h(T) − (1/N)log(#C)| < ϵ_{n}, and

• for 1 ≤ k < n,

\frac{1}{N} logN((P_{n})_{0}^{N−1}|(P_{k}^{∗})_{0}^{N−1}) < h(T |P_{k}^{∗}) + ϵ_{n} .

Then define S_{n} = S(P, C, N).

Next we define the bonding map π: S_{n} → S_{n−1} (which typically will not be surjective). To do this, for each of the words W = W (i) of length N used to define S_{n} = S(P, C, N) as in Part I, we will define an S_{n−1} word W′ of length N, and set (πx)[j, j + N − 1] = W′ whenever x[j, j + N − 1] = W. So, consider W = C_{i}00 … 0, and recall C_{i} = ⋂_{j=0}^{N−1} T^{−j}P(i, j) . Let K = N/(N_{n−1}). Using the refinement condition, for 0 ≤ k < K pick C_{I(k)} ∈ C_{n−1} such that P (i, kN_{n−1}) ⊂ C_{I(k)}. Let V_{k} denote the word which is the symbol C_{I(k)} followed by N_{n−1} − 1 zeros. Then set W′ = V_{0}V_{1} ⋯ V_{K−1}.

This mapping rule on words gives a well defined map π: S_{n} → S_{n−1} because if a concatenation of W ’s corresponds to a nonempty set B in T, then the corresponding concatenation of W′s corresponds to a set which contains B, and is therefore nonempty. For any x = (x^{(1)}, x^{(2)}, …) ∈ S, we have K(x^{(1)}) ⊃ K(x^{(2)}) ⊃ …, with the diameters of the K(x^{(n)}) going to zero (because the mesh of P_{n} goes to zero). So, the rule x ↦ ∩_{n}K(x^{(n)}) gives a well defined map φ from S to T. The map φ is surjective by a compactness argument because for each n, the union of the sets K(x^{(n)}) is all of T. The map φ is obviously equivariant.

Part III. It remains to check the entropy claims. Because φ: S ↠ T is surjective, h(T) ≤ h(S). On the other hand, clearly

h(S) ≤ \overline{lim}_{n} h(S_{n}) ≤ lim_{n} (h(T) + ϵ_{n}) = h(T) .

So it remains to verify e^{∗}(φ) ≤ h^{∗}(T). We will check that lim h(S|φ^{−1}P_{k}) ≤ h^{∗}(T). So, fix P_{k}, fix n > k and let N = N_{n}. Suppose for elements P_{i} of P_{k} that

U = ⋂_{i=0}^{N−1}(φ^{−1}T^{−i}P_{i}) ∈ ⋁_{i=0}^{N−1} S^{−i}(φ^{−1}P_{k}) .

Define the set of words

E = {y_{0}y_{1} … y_{N−1} : ∃x ∈ U, y = x^{(n)}, y_{0} ≠ 0} .

Suppose we have the following CLAIM:

\frac{1}{N} log#E ≤ h(T |P_{k}^{∗}) + ϵ_{n} .

Let Q_{n} be the open cover/partition of S according to x_{0}^{(n)}. It follows from the claim that

h(S, Q_{n}|φ^{−1}P_{k}) = inf_{M} \frac{1}{M} logN((Q_{n})_{0}^{M−1}|(φ^{−1}P_{k})_{0}^{M−1})
≤ h(T |P_{k}^{∗}) + ϵ_{n}

and consequently (because the mesh of P_{k}^{∗} goes to zero)

e^{∗}(φ) = lim_{k} h(S|φ^{−1}P_{k}) = lim_{k} lim_{n} h(S, Q_{n}|φ^{−1}P_{k})

≤ lim_{k} h(T |P_{k}^{∗}) = h^{∗}(T) .

It remains then to prove the Claim. So suppose x ∈ U, w = x^{(k)} and y = x^{(n)}. We have associated sequences \overline{w} and \overline{y} on symbols from P_{k} and P_{n} respectively. Because x ∈ U, for 0 ≤ i < N the closure of the set \overline{w}_{i} must intersect P_{i}, so the open set \overline{w}_{i} must intersect the open set P_{i}, and \overline{w}_{i} ⊂ P_{i}^{∗}. Then because P_{n} refines P_{k}, the set ∩_{i=0}^{N−1}T^{−i}P_{i}^{∗} contains the set ∩_{i=0}^{N−1}T^{−i}\overline{y}_{i}. Now the cardinality of E cannot exceed N((P_{n})_{0}^{N−1})|(P_{k}^{∗})_{0}^{N−1}), because if it did, we could replace in C_{n} the subcollection of elements contained in ∩_{i=0}^{N−1}T^{−i}P_{i}^{∗} with a smaller subcollection covering ∩_{i=0}^{N−1}T^{−i}P_{i}^{∗}, and thus contradict the choice of C_{n} as a minimum cardinality cover. Consequently we have

#E ≤ N((P_{n})_{0}^{N−1}|(P_{k}^{∗})_{0}^{N−1})

and now the Claim follows from the construction of S_{n}. □

Corollary A.2. Suppose T is asymptotically h-expansive. Then there is an asymptotically h-expansive zero-dimensional system S and a quotient map φ: S ↠ T such that h(S) = h(T) and e^{∗}(φ) = 0.

Proof. By the previous result, we have φ: S ↠ T with S zero dimensional such that h(S) = h(T) and h^{∗}(S) ≤ h^{∗}(T) = 0. □

Remark A.3. For φ: S → T, recall from Facts 5.6 that e^{∗}(φ) ≥ \frac{1}{2}h^{∗}(T). So in the corollary above, the assumption h^{∗}(T) = 0 is necessary for e^{∗}(φ) = 0.

Remark A.4. It is not possible without further hypotheses to add to the conclusion of Theorem A.1 the requirement h^{∗}(S) = 0. This is because an asymptotically h-expansive system can be covered by a subshift, but there are T of finite entropy which cannot be covered by a subshift.

Question A.5. Is every system T covered by an equal entropy zero dimensional system of equal residual entropy?

We see the last question does have an affirmative answer when T is asymptotically h-expansive or (from the next section) finite dimensional.

Appendix B. Zero dimensional covers of finite dimensional systems

If T is a finite dimensional system (that is, its domain has finite covering dimension), then there are very strong results on the existence of covers φ: S ↠ T, with S zero dimensional and φ giving a good approximation of T by S. We will state two theorems, and then explain how they follow from the work of Kulesza [Ku1] and Thomsen[T1, T2, T3].

Theorem B.1. Suppose T is finite dimensional and the set of periodic points of T is zero dimensional. Then there is a zero dimensional system S and a quotient map φ: S ↠ T such that the following hold.

(1) φ is at most (n + 1)^{n} to one.

(2) φ is residually one-to-one.

(3) φ has defect zero.

(4) ρ(S) = ρ(T).

That φ is residually one-to-one means that there are residual (second category) sets in S and T such that restriction of φ gives a bijection between these sets. The meaning of “defect zero” is explained below.

Theorem B.2. Suppose T is finite dimensional. Then there is a zero dimensional system S and a quotient map φ: S → T such that the following hold.

(1) For every S-invariant Borel probability μ, h_{μ}(S) = h_{φ_{∗}μ}(T).

(2) For every subsystem R of T, h(R) = h(φ^{−1}R).
(3) ρ(S) = ρ(T).

The condition 1 in Theorem B.2 is the condition Ledrappier [Le] used to define S as a principal extension of T. Let us note that this condition, and even more so the condition of Theorem B.3 below, become particularly subtle to arrange when there are uncountably many T -invariant ergodic Borel probabilities. In this case, when constructing regular closed partitions, itineraries through which will generate symbolic sequences for S, one cannot easily perturb partition boundaries to null sets for all measures of interest.

For φ: S ↠ T with S zero dimensional, we now recall Thomsen’s definition [T2] of the defect D(φ) of the factor map φ. (Thomsen considered systems which are not necessarily homeomorphisms; but here as in the rest of this paper we consider only homeomorphisms.) The definition has several layers. Given a finite collection F = {F_{i} : i ∈ I} of subsets of T, we set

q_{k}(x, F) = #{(i_{1}, i_{2}, …, i_{k}) : x ∈ ⋂_{j=1}^{k} T^{−j+1}(F_{i_{j}})}

for all x ∈ T, k ∈ ℕ, and then

q_{k}(T, F) = max_{x∈T} q_{k}(x, F)

and

Q(T, F) = lim_{n} \frac{1}{n}logq_{n}(T, F) .

Then we define the defect of φ as

D(φ) = sup_{P} Q(T, φ(P))

where the supremum is over all clopen partitions of S (i.e. partitions of S into disjoint nonempty closed open sets). For example, D(φ) = 0 if φ is bijective. There is an easy but important observation (Lemma 6.6 of [T1]): if P_{1} and P_{2} are clopen partitions and P_{2} refines P_{1}, then Q(T, φ(P_{1})) ≤ Q(T, φ(P_{2})). The meaning of the defect is captured by Thomsen’s Defect Variational Principle:

Theorem B.3. [T3] Suppose φ: S ↠ T and S is zero dimensional. Then

D(φ) = sup_{μ} ∫ log#φ^{−1}(x) dμ(x),

where the supremum is over all T -invariant Borel probability measures (or equivalently over all T -invariant ergodic Borel probability measures).

Let us consider the relation of D(φ) and e^{∗}(φ). Clearly all values of D(φ) are compatible with e^{∗}(φ) = 0. On the other hand, if φ = Id_{T}, then D(φ) = 0 but e^{∗}(φ) = h^{∗}(T). In the case that T is asymptotically h-expansive, we have the following result.

Proposition B.4. Suppose φ: S ↠ T, S is zero dimensional, h^{∗}(T) = 0 and D(φ) is finite. Then e^{∗}(φ) = 0.

Proof. If D(φ) is finite, then it follows from the Defect Variational Principle that φ is a principal extension. From the Variational Principle for Quotient Maps (6.7), we then conclude that e^{∗}(φ) = 0. □

A zero dimensional extension φ: S ↠ T can be used to study periodic points or invariant measures for T by studying periodic points or invariant measures for S. This approach requires a reasonable correspondence under φ of these objects. If the extension φ has defect zero, then the correspondence is close as possible.

Facts B.5. Suppose S is zero dimensional and φ : S ↠ T has defect zero. Then the following hold.

(1) h_{μ}(S) = h_{φ_{∗}μ}(T), for every S-invariant Borel probability.

(2) Every T -periodic point has a unique φ preimage.

(3) For every positive integer n, #{x : S^{n}x = x} = #{x : T^{n}x = x} .

(4) For every subsystem R of T, h(φ^{−1}R) = h(R).

The first three facts are obvious from the Defect Variational Principle, and the fourth follows by also applying the usual variational principle.

Thomsen defined a zero dimensional extension φ: S ↠ T to be perfect when φ is bounded to one with D(φ) = 0, and constructed perfect extensions for several classes of systems T. He also introducted the logarithmic covering dimension of T [T1], and showed this vanishes if and only there is a zero dimensional extension φ: S ↠ T with defect zero. Positive dimension of the set of periodic points is an obstruction to existence of a defect zero extension [T1].

Next we recall a theorem of Kulesza.

Theorem B.6. [Ku1] Suppose T is a self homeomorphism of a compact metrizable space of dimension n < ∞, and the periodic point set of T is zero-dimensional. Then there is a self homeomorphism S of a zero dimensional compact metrizable space, and a quotient map φ: S ↠ T such that no point of T has more than (n+1)^{n} preimages.

The statement of the theorem is false without the hypothesis on the periodic point set [Ku1]. However, the bound (n + 1)^{n} can be improved to (n + 1)[Ku2], which of course is best possible [HW].

Before proving our two theorems, we isolate a lemma.

Lemma B.7. Suppose φ: S ↠ T, S is zero dimensional and φ is uniformly finite to one. Then ρ(S) = ρ(T).

Proof. Clearly h(S) = h(T) and every subshift cover of S is a subshift cover of T, so it suffices to show, given a subshift S′ and γ: S′ ↠ T, that there is some subshift cover S′′ of S such that h(S′′) = h(S′). For this, let F be the fibered product of S′ and S by the maps γ and φ. That is, F is the subset of S′ × S on the points (x, y) such that γ(x) = φ(y), and the projection map (x, y) ↦ y maps F onto S. The projection p: F ↠ S′ (given by (x, y) ↦ x) is uniformly finite to one, so h(F) = h(S′) and SM(p) = 0. Because F is zero dimensional and the subshift S′ has conditional topological entropy zero, it follows from Lemma 6.2 (or the full Variational Principle 6.7) that e^{∗}(p) = 0. Then Facts 5.6(5) gives h^{∗}(F) = 0, and F is asymptotically h-expansive. It follows from Theorem 7.4 that there exists a quotient map β: S′′ ↠ F such that S′′ is a subshift and h(S′′) = h(F). Then we have S′′ ↠ F ↠ S and h(S′′) = h(F) = h(S′) as required. □

Now we can prove our two theorems.

Proof of Theorem B.1. Theorem B.6 is the main result of [Ku1] (and from this and the last lemma it follows that ρ(S) = ρ(T)). Examining the map φ which Kulesza constructed, we see it is residually one-to-one (this is easy) and has defect zero. We will describe a little of his construction to indicate why D(φ) = 0.

Kulesza constructed a certain sequence of closed regular covers D_{i}. The covering zero dimensional system S can be viewed as the inverse limit of subshifts S_{i} on alphabets D_{i}. In this construction, for any given clopen partition P there is some i such that P is refined by the time-zero partition P_{i} for S_{i}. So, φ will have defect zero if for each i, Q(T, φ(P_{i})) = 0. Here φ(P_{i}) is the cover D_{i}, and the conclusion will follow if for some positive integer M_{i}, we have for every x in T and k ∈ ℕ that q_{k}(x, D_{i}) ≤ M_{i}.

The cover D_{i} is defined as

D_{i} = ⋁_{−i≤h≤i} (⋁_{j≤i} T^{h}(C_{j}))

where the C_{j} are finite regular closed covers constructed by Kulesza such that for every x,

#{m ∈ ℤ : T^{m}x ∈ ⋃_{j>0} bd(C_{j})} ≤ (n + 1)^{n} .

Here “regular closed”means that each element of the cover is the closure of its interior, and distinct elements have disjoint interiors. It follows that if a point x is contained in more than one element of the cover T^{−m}D_{i}, then T^{m+h}x must lie in one of the sets bdC_{j}, for some integer h such that −i ≤ h ≤ i. Thus for all x,

q_{k}(x, D_{i}) ≤ [(n + 1)^{n}]^{2i+1}

and this shows D(φ) = 0. □

Proof of Theorem B.2. Let Z be a zero entropy subshift without periodic points. Now T × Z has no periodic points, so by Kulesza’s theorem there is a quotient map ψ: S ↠ T × Z such that S is zero dimensional and ψ is uniformly finite to one. Let π be the projection T × Z ↠ T and let φ = πψ. Clearly ρ(T) = ρ(T × Z). By Lemma B.7, ρ(T × Z) = ρ(S), so ρ(T) = ρ(S). The straightforward verification of the other claims is left to the reader. □

Remark B.8. There is no general inequality between ρ(T) and the minimum defect of a quotient map from a zero dimensional space onto T. If T is zero dimensional, then the minimum defect is obviously zero, but ρ(T) is arbitary in [0, +∞]. On the other hand, if T is the identity map on a compact metrizable space of dimension n ∈ {0, 1, …, ∞}, then the minimum defect is log(n + 1) but ρ(T) = 0.

Remark B.9. In the special case that T is expansive, it is a simple consequence of uniform continuity that any zero dimensional extension S ↠ T factors through some subshift extension, S ↠ S′ ↠ T. Then the defect of S′ ↠ T is at most that of S ↠ T. For T is expansive, it is well known that dim(T) is finite [Ma1] and Per(T) is countable ([DGS], Prop. 16.10). Consequently we have the following corollary of Theorem B.1.

Corollary B.10. If T is an expansive homeomorphism, then there is a cover φ: S ↠ T such that S is a subshift and φ has defect zero.

Appendix C. Infinite residual entropy on a surface

The purpose of this appendix is to construct a selfhomeomorphism T of the unit disc 𝔻 = {z ∈ ℂ: |z| ≤ 1} such that T has finite entropy and infinite residual entropy. T will be the identity on the boundary of 𝔻, so this example can be realized on any surface. Parts of the construction can be done smoothly and parts more generally. We thank Mike Handel, Judy Kennedy, Mike Shub and John Smillie for helpful consultations.

Fix a homeomorphism T_{0} : 𝔻 → 𝔻 with the following properties:

• T_{0} has finite entropy

• T_{0} is C^{1} with det(DT_{0}) > 0

• T_{0} = Id on the boundary of 𝔻

• there is a subset E of int(𝔻) such that T_{0}|_{E} is a mixing SFT S of entropy

log λ > 0. (We do not have an explicit reference for the existence of such a T_{0}, but it is not difficult to construct T_{0} (with λ = 2) by suitably extending Smale’s horseshoe construction ([S], pp. 772-773) to a disc diffeomorphism.) Fix α such that 1 < α < λ. Below, D(q, ϵ) represents a closed disc of radius ϵ centered at q, also we use notation for annuli such as the following: [a < |z − q_{n}| ≤ b] represents {z : a < |z − q_{n}| ≤ b}. Also, P_{n}^{0}(S) denotes the set of points in S-orbits of cardinality n.

Lemma C.1. There is a collection of pairwise disjoint discs D(q_{k}, ϵ_{k}) contained in the interior of 𝔻 such that

• the points q_{k} are periodic points of S

• lim sup \frac{1}{n}log|Q_{n}| ≥ logα, where Q_{n} = \{q_{k}: k ∈ ℕ, q_{k} ∈ P_{n}^{0}(S)\}

• the set Q = ∪Q_{n} is invariant

• ϵ_{k} is the same number ϵ(n) for all q_{k} ∈ |P_{n}^{0}(S)|.

Proof. Choose N such that |P_{n}^{0}(S)| ≥ α^{n} for all n ≥ N (here |P_{n}^{0}(S)| denotes the set of points in S-orbits of cardinality n). This is possible because S is a mixing SFT and logα < h(S).

To prove the lemma, it suffices to choose recursively, for n = N, N + 1, … ((i)) a mixing SFT S^{(n)} ⊂ S such that h(S^{(n)}) > logα,

|P_{k}^{0}(S^{(n)})| ≥ α^{k} if k > n ,
and S^{(n)} ⊂ S^{(n−1)} if n > N ;

((ii)) a set Q_{n} ⊂ P_{n}^{0}(S) such that |Q_{n}| ≥ α^{n} ; and ((iii)) ϵ(n) > 0 such that the family of discs D(q, ϵ(k)), with q ∈ Q_{k} and N ≤

k ≤ n, are pairwise disjoint and disjoint from S^{(n)}, and are contained in the interior of 𝔻.

We begin with n = N. Define Q_{N} = P_{N}^{0}(S). Pick a mixing SFT S′ such that logα < h(S′) < h(S) and S′ has a fixed point (e.g. using [Kr1]). Pick N_{1} > N such that |P_{k}^{0}(S)| ≥ |P_{k}^{0}(S′)| ≥ α^{k} for k ≥ N_{1}. Using the Covering Lemma 2.1 of [B1], produce a mixing SFT S′′ such that h(S′′) = h(S′) and

|P_{k}^{0}(S′′)| = 0 ,  k ≤ N ,
|P_{k}^{0}(S′′)| = |P_{k}^{0}(S)| ,  N < k < N_{1} ,
|P_{k}^{0}(S′)| ≤ |P_{k}^{0}(S′′)| ≤ |P_{k}^{0}(S)| ,  k ≥ N_{1} .

By Krieger’s Embedding Theorem [Kr2], we may assume S′′ ⊂ S (and necessarily then, Q_{N} and S′′ are disjoint). Now set S^{(N)} = S′′ and choose ϵ(N) to satisfy (iii) for n = N.

The recursive step is much the same. Suppose n + 1 > N and we have carried out the choices above for N, …, n. Define Q_{n+1} = P_{n+1}^{0}(S^{(n)}) (so, |Q_{n+1}| ≥ α^{n+1}, and Q_{n+1} as a subset of S^{(n)} is disjoint from the discs previously chosen for k ≤ n). Pick a mixing SFT S^{∗} such that logα < h(S^{∗}) < h(S^{(n)}) and S^{∗} has a fixed point. Pick N_{2} > n + 1 such that |P_{k}^{0}(S^{(n)})| ≥ |P_{k}^{0}(S^{∗})| ≥ α^{k} if k ≥ N_{2}. Then pick a mixing SFT S^{∗∗} such that h(S^{∗∗}) = h(S^{∗}) and

|P_{k}^{0}(S^{∗∗})| = 0 ,  k ≤ n + 1 ,
|P_{k}^{0}(S^{∗∗})| = |P_{k}^{0}(S^{(n)})| ,  n + 1 < k < N_{2} ,
|P_{k}^{0}(S^{∗})| ≤ |P_{k}^{0}(S^{∗∗})| ≤ |P_{k}^{0}(S^{(n)})| ,  k ≥ N_{2} .

As before, by Krieger’s Embedding Theorem we may assume S^{∗∗} ⊂ S^{(n)}. Then define S^{(n+1)} = S^{∗∗}. Then choose ϵ(n + 1) to satisfy (iii).

This finishes the lemma. □

The map T will be the uniform limit of homeomorphisms T_{n}: 𝔻 → 𝔻. First we describe how T_{1} is constructed as a modification of T_{0}. Fix a choice of discs D_{n} = D(q_{n}, ϵ_{n}) satisfying the statement of Lemma C.1. Define a map π_{0} : 𝔻 → 𝔻 by setting

π_{0}(z) = z ,  if z ∉ ⋃_{n=1}^{∞} D_{n}
π_{0}(q_{n} + z) = q_{n} ,  if |z| ≤ \frac{ϵ_{n}}{2}
π_{0}(q_{n} + z) = q_{n} + \frac{2}{ϵ_{n}} (|z| − \frac{ϵ_{n}}{2} )z ,  if \frac{ϵ_{n}}{2} < |z| ≤ ϵ_{n} .

The map π_{0} maps the half-open annulus D_{n} \ D(q_{n}, ϵ_{n}/2) radially and homeomorphically to the punctured disc D_{n} \ {q_{n}}. The restriction of π_{0} to D \ π_{0}^{−1}Q is a homeomorphism onto its image. For z ∈ 𝔻 \ π_{0}^{−1}Q, define T_{1}(z) = π_{0}^{−1}T_{0}π_{0}(z). It remains to define T_{1} on the discs D(q_{n}, ϵ_{n}/2).

Suppose T_{0}(q_{n}) = q_{k}. Because T_{0} is differentiable and nonsingular at q_{n}, the map T_{1} defined so far on [ϵ_{n}/2 < |z − q_{n}| ≤ ϵ_{n}] extends continuously to a map

β_{n}: [|z − q_{n}| = \frac{ϵ_{n}}{2} ] ↦ [|z − q_{k}| = \frac{ϵ_{k}}{2} ] .

Because ϵ_{k} = ϵ_{n} and det(DT_{0}) > 0 at q_{n}, there is an orientation preserving homeomorphism h_{n}: (ϵ_{n}/2)S^{1} → (ϵ_{n}/2)S^{1} such that

β_{n}: q_{n} + z ↦ q_{k} + h_{n}(z) , if |z| = \frac{ϵ_{n}}{2} .

Because an orientation preserving circle homeomorphism is isotopic to the identity, there is a homeomorphism

H_{n}: [\frac{ϵ_{n}}{4} ≤ |z| ≤ \frac{ϵ_{n}}{2} ] → [\frac{ϵ_{n}}{4} ≤ |z| ≤ \frac{ϵ_{n}}{2} ]

such that H_{n}(z) = h_{n}(z) for |z| = ϵ_{n}/2 and H_{n}(z) = z for |z| = ϵ_{n}/4. For ϵ_{n}/4 ≤ |z| ≤ ϵ_{n}/2, we define T_{1}(q_{n} + z) = q_{k} + H_{n}(z). Finally, for |z| ≤ ϵ_{n}/4, we define

T_{1} : q_{n} + z ↦ q_{k} + \frac{ϵ_{n}}{4} T_{0}(4z/ϵ_{n}) .

So, T_{1} defines a map D(q_{n}, ϵ_{n}/4) → D(q_{k}, ϵ_{k}/4) which is a miniature copy of T_{0}. This completes the definition of T_{1}. The map T_{1} : 𝔻 → 𝔻 is a homeomorphism. If m is the period of q_{n}, then the union of the discs D((T_{0})^{i}q_{n}, \frac{1}{4}ϵ_{n}), 0 ≤ i < m, is T_{1}-invariant; and the restriction of T_{1}^{m} to D(q_{n}, ϵ_{n}/4) is topologically conjugate to T_{0}^{m}. The map π_{0} : 𝔻 → 𝔻 is a semiconjugacy from T_{1} to T_{0}, and π_{0} = Id on the complement of ∪D_{n}.

To define T_{2} as a modification of T_{1}, we repeat the “blow and sew”process above inside each of those discs D_{k} = D(q_{k}, \frac{1}{4}ϵ_{k}). Each D_{k} contains disjoint discs

D_{k,n} = D(q_{k} + \frac{1}{4}ϵ_{k}q_{n}, \frac{1}{4}ϵ_{k}ϵ_{n}), 1 ≤ n < ∞ ,

centered at T_{1}-periodic points, and these points are blown up into discs into each of which we sew a miniature copy of T_{0}. Just as we defined π_{0} : 𝔻 → 𝔻 in defining T_{1} as a modification of T_{0}, we define π_{1;k}: D_{k} → D_{k} in defining T_{2} as a modification of T_{1}, in particular the map π_{1,k} is the identity map on the complement of ∪_{n}D_{k,n}. We define π_{1} : 𝔻 → 𝔻 by setting π_{1} = π_{1;k} on D_{k} and π_{1} = Id elsewhere. The map π_{1} is a semiconjugacy from T_{2} to T_{1}, and π_{1} = Id on the complement of ∪_{k,n}D_{k,n}, which is a subset of ∪_{n}D_{n}. Note, π_{1}π_{0} = π_{0}.

Recursively, to define T_{n+1} as a modification of T_{n}, we “blow and sew”in each of a family of disjoint discs D_{k(1),...,k(n)}, blowing up certain T_{n}-periodic points to discs and sewing in miniature copies of T_{0}. Let Q^{(n)} denote the set of T_{n}-periodic points blown up into discs in the definition of T_{n+1} as a modification of T_{n}. In the process of constructing T_{n+1} from T_{n}, we obtain a semiconjugacy π_{n} from T_{n+1} to T_{n}, where π_{n} maps each D_{k(1),...,k(n)} onto itself and is the identity elsewhere. We have π_{n}π_{n−1} = π_{n−1}.

For n ≥ 1, let G_{n} denote the union of the disks D_{k(1),...,k(n)}. The sets G_{n} are nested and T_{n} = T_{n−1} outside G_{n}. So for all z and all m > 0, dist(T_{n}(z), T_{n+m}(z)) cannot be more than the maximum diameter of a disc D_{k(1),...,k(n+1)}, which goes to zero uniformly with n. Therefore the T_{n} converge uniformly to a continuous map T: 𝔻 → 𝔻.

Next we will observe that T is topologically conjugate to the homeomorphism T′ defined as the inverse limit of the maps T_{n}, with bonding maps π_{n}: T_{n+1} ↠ T_{n}. To show T is conjugate to T′, it suffices to produce semiconjugacies φ_{n}: T ↠ T_{n} such that φ_{n} = π_{n}φ_{n+1} and such that ∏_{n=1}^{∞} φ_{n} is injective on 𝔻. Simply define φ_{n} = π_{n}, then φ_{n} = π_{n}φ_{n+1}. The images under π_{n} of G_{n+1} and its complement are disjoint, and the map π_{n} is one-one on the complement of G_{n+1}. Consequently φ is injective on the complement of ∩_{n}G_{n}. However, φ is injective on ∩_{n}G_{n} as well, because each nested sequence of discs

D_{k(1)} ⊃ D_{k(1),k(2)} ⊃ D_{k(1),k(2),k(3)} ⊃ ⋯

shrinks to a single point, and for each n, each level-n disc D_{k(1),...,k(n)} is π_{n−1}- invariant.

With this inverse limit presentation for T, it is a straightforward matter to see that the argument for infinite residual entropy in Section 3 adapts to show that T has infinite residual entropy.

For each n, under the semiconjugacy π_{n}: T_{n+1} ↠ T_{n} there are some periodic orbits of T_{n} whose inverse images are subsystems of entropy h(T_{0}), and the restriction of π_{n} to the complement of the union of these inverse images is bijective. It follows by induction on n that there is no ergodic T_{n+1}-invariant Borel probability μ with h_{μ}(T_{n+1}) > h(T_{0}); so, by the variational principle and ergodic decomposition, we have h(T_{n+1}) ≤ h(T_{0}). Thus each h(T_{n}) = h(T_{0}) and therefore the inverse limit system T satisfies h(T) = h(T_{0}) < ∞. This completes the example.

Remarks C.2. We only needed DT_{0} existing and positive at the disc centers q_{n}. Also, if in the construction we begin with T_{0} a C^{2} map, then with some modifications to the “blow and sew”operation (and some unpleasantly technical additional arguments), we can arrange each T_{n} to be C^{1}, and T to be differentiable on the complement of ∩_{n}G_{n}. But we see no way to modify the construction to achieve differentiability of T on ∩_{n}G_{n}: at a point in this Cantor set, the local picture need not approach a linear map as the scale shrinks; for example on a sequence of scales the map could be locally approximated by different linear maps.

The effort to construct the example above naturally raises a technical question:

Question C.3. Suppose T is a homeomorphism of a Cantor set C contained in the interior of a disc D, and T has finite entropy. Does T extend to a finite entropy homeomorphism of D?

It is a well known consequence of the Schoenflies Theorem that any homeomorphism of C above extends to a homeomorphism of D which is the identity on the boundary. If the entropy of the extension can be controlled, then one has a general method for constructing finite entropy, infinite residual entropy homeomorphisms of a surface.

Appendix D. Intermediate residual entropy

We will prove the following result.

Theorem D.1. Suppose 0 < a < ∞ and 0 ≤ b ≤ ∞. Then there is a zero dimensional system T with h(T) = a and ρ(T) = b.

Remark D.2. These results are also in [Do2]; the constructions are very different. Note, if h(T) = 0 then ρ(T) = 0 (Cor.7.6 or [Do2]), and if h(T) = ∞ then ρ(T) = ∞. So Theorem D.1 covers all the possible cases.

The heart of the proof of Theorem D.1 is the explicit construction proving Proposition D.5 below. The rest of the proof rests on the following two lemmas. We use the notation that T_{(n)} denotes the discrete tower of height n built over the transformation T. Explicitly, if X is the domain of T, then X × {1, 2, …, n} is the domain of T_{(n)}, which maps by the rule

T_{(n)} : (x, i) ↦ (x, i + 1)  if i ≠ n
↦ (T x, 1)  if i = n .

It is well known and easy to see that h(T_{(n)}) = (1/n)h(T).

Lemma D.3. Suppose T is a selfhomeomorphism of a compact metric space. Then (1) ρ(T^{n}) = nρ(T).

(2) ρ(T_{(n)}) = (1/n)ρ(T).

Proof. (1) If S is a subshift cover of T, then S^{n} is a subshift cover of T^{n}, and h(S^{n}) − h(T^{n}) = n[h(S) − h(T)]. Thus ρ(T^{n}) ≤ inf_{S} n[h(S) − h(T)] = nρ(T).

Conversely, if S is a subshift and φ: S ↠ T^{n}, then S_{(n)} is a subshift cover of T by the map (x, i) ↦ T^{i−1}φx if 1 ≤ i ≤ n. Therefore ρ(T) ≤ h(S_{(n)}) − h(T) = (1/n)[h(S) − h(T^{n})] and we obtain ρ(T) ≤ inf_{S}(1/n)[h(S) − h(T^{n})] = (1/n)ρ(T^{n}).

(2) The system (T_{(n)})^{n} is the disjoint union of n copies of T, so ρ((T_{(n)})^{n}) = ρ(T). Then it follows from (1) that ρ(T_{(n)}) = (1/n)ρ(T).

□

Given a sequence T_{n} of systems, we let (T_{n})_{∞} denote the one point compactification system (in which the added point is a fixed point). In the lemma below we regard the systems in such a sequence T_{n} as being disjoint.

Lemma D.4. Let T = (T_{n})_{∞}. Define • α = inf{h(S): S is a subshift and S ↠ T } ,

• α_{n} = inf{h(S): S is a subshift and S ↠ T_{n}} .

Then α = sup α_{n}.

Proof. Clearly α ≥ sup α_{n}. So suppose sup α_{n} < ∞ and ϵ > 0. Pick a mixing SFT U such that 0 < h(U) − sup α_{n} < ϵ. For each n, pick a subshift cover S_{n} of T_{n}, with h(S_{n}) < h(U), and pick a mixing SFT R_{n} with a fixed point such that S_{n} ⊂ R_{n} and h(R_{n}) < h(U). Let Z be the identity map on the space which is the convergent sequence {0} ∪ {1/n: n ∈ ℕ}. Using Cor. 7.6, pick some zero entropy subshift W such that U × W ↠ U × Z. Recall that U ↠ R_{n} for each n by [B1], so U × Z ↠ (R_{n})_{∞}. Putting all this together, we see

U × Z ↠ (R_{n})_{∞} ⊃ (S_{n})_{∞} ↠ (T_{n})_{∞} = T .

Taking the inverse image of T inside U × Z, we get a subshift V such that h(V) ≤ h(U × Z) = h(U) < sup α_{n} + ϵ. □

Proof of Theorem D.1. Suppose 0 < a, b < ∞. Given m ∈ ℕ, pick positive integers k, n such that

(a + b) − \frac{1}{m} < \frac{k}{n}(log2) < a + b ,  and
\frac{1}{n}(log2) < a .

Using Proposition D.5 below, given ϵ > 0 we may pick T such that

log2 ≤ h(T) ≤ log2 + ϵ ,  and
(klog2) − ϵ ≤ ρ(T) + log2 ≤ klog2 .

Then

\frac{1}{n}(log2) ≤ h(T_{(n)}) ≤ \frac{1}{n}[(log2) + ϵ] ,  and
\frac{k}{n}(log2) − \frac{ϵ}{n} ≤ ρ(T_{(n)}) + h(T_{(n)}) ≤ \frac{k}{n}(log2) + \frac{ϵ}{n} ,

so for small enough ϵ we have

h(T_{(n)}) < a , and
(a + b) − \frac{1}{m} < ρ(T_{(n)}) + h(T_{(n)}) < (a + b) .

Choose S_{m} to be a system T_{(n)} satisfying the last two lines. Let S_{0} be a subshift such that h(S_{0}) = a. Regard S_{0}, S_{1}, S_{2}, … as pairwise disjoint. Let S be the one point compactification of the systems S_{0}, S_{1}, S_{2}, …. Then h(S) = a, and ρ(S) = b as a consequence of Lemma D.4. This finishes the case 0 < b < ∞.

If b = ∞, then again we may take S to be the one point compactification of systems S_{0}, S_{1}, S_{2}, …, with h(S_{0}) = a and h(S_{n}) < a for n > 0, but in this case we require ρ(S_{n}) → ∞. For the case b = 0, let S = S_{0}.

□

The rest of the section is devoted to the following result, which is the heart of the matter.

Proposition D.5. Suppose ε > 0 and r ∈ ℕ. Then there is an inverse limit T of mixing sofic shifts T_{n} and surjective bonding maps π_{n}: T_{n+1} → T_{n} such that

• T is a quotient of the full shift on 2^{r+1} symbols (by construction).

• log2 ≤ h(T) ≤ log2 + ε.

• h(R) ≥ log(2^{r+1}) for every subshift R such that T is a quotient of R.

Thus we will have rlog2 − ε ≤ ρ(T) ≤ rlog2.

We prepare for the definition of T. Fix r ∈ ℕ. Define an ordering ≺ on the set ℕ ∪ ℕ^{2} ∪ ⋯ ∪ ℕ^{r} as follows. Let <_{lex} denote lexicographic ordering on

ℕ∪ℕ^{2}∪⋯∪ℕ^{r}. For (n_{1}, …, n_{i}) ∈ ℕ∪ℕ^{2}∪⋯∪ℕ^{r}, let ||(n_{1}, …, n_{i})|| := ∑_{1≤k≤i} n_{k}.

Let n, m ∈ ℕ∪ℕ^{2}∪⋯∪ℕ^{r}. Then define n ≺ m if (||n|| < ||m||) or (||n|| = ||m|| and n <_{lex} m). We will use just two properties of this order: when r = 1, this is the usual order on ℕ; and for r ≥ 2 it holds that (n_{1}, …, n_{k−1}) ≺ (n_{1}, …, n_{k}), if 2 ≤ k ≤ r. Let i: ℕ → ℕ ∪ ℕ^{2} ∪ ⋯ ∪ ℕ^{r} denote the bijection such that j < k ⇒ i(j) ≺ i(k).

Let S = {0, 1}^{ℤ} denote the full 2-shift. Fix an enumeration of the finite S-orbits, say O_{1}, O_{2}, . . .. Fix ε > 0. Fix a sequence (N_{n})_{n≥1} of natural numbers.

We define f_{0} : S^{r+1} → S to be the projection onto the first coordinate, f_{0}(x^{1}, …, x^{r+1}) := x^{1}. For n ≥ 1 we define f_{n}: S^{r+1} → S as follows. We have that i(n) = (k_{1}, …, k_{s}) for some 1 ≤ s ≤ r. Let f_{n}(x^{1}, …, x^{r+1})_{0} := x_{0}^{s+1} if O_{k_{i}} ∩ [x_{−N_{n}}^{i}, …, x_{N_{n}}^{i}]_{N_{n}} ≠ ∅ for all 1 ≤ i ≤ s, otherwise f_{n}(x^{1}, …, x^{r+1})_{0} := 0. Then each f_{n} is a shift invariant, continuous onto map. For n ∈ ℕ, let T_{n} := (f_{0}, …, f_{n−1})(S^{r+1}). Thus T_{n} ⊂ S^{n}. Let π_{n}: T_{n+1} → T_{n} denote the projection onto the first n coordinates. This defines an inverse limit T which is a quotient of S^{r+1}. Note that T_{1} = f_{0}(S^{r+1}) = S, thus h(T) ≥ log2.

We shall now show that there is a choice of the sequence N_{n} such that h(T) ≤ log2 + ε. For each orbit O_{n} fix a point p_{n} ∈ O_{n}. For n ∈ ℕ let i(n) = (k_{1}, …, k_{s}). Define a map h_{n}: S^{r+1} → (∏_{1≤i≤s}{1, …, |O_{k_{i}}|} ∪ {0})^{ℤ} as follows. Let h_{n}(x^{1}, …, x^{r+1})_{0} := (m(1), …, m(s)) if x^{i}[−N_{n}, N_{n}] = S^{m(i)}p_{k_{i}}[−N_{n}, N_{n}] for all 1 ≤ i ≤ s and let h_{n}(x^{1}, …, x^{r+1})_{0} := 0 else. Note, h_{n}(x^{1}, …, x^{r+1})_{0} ≠ 0 implies f_{n}(x^{1}, …, x^{r+1})_{0} = x_{0}^{s+1}.

Note also, h_{n}(x^{1}, …, x^{r+1})_{0} = (m(1), …, m(s)) implies h_{n}(x^{1}, …, x^{r+1})_{1} = 0 or h_{n}(x^{1}, …, x^{r+1})_{1} = (m(1) + 1 mod |O_{k_{1}}|, …, m(s) + 1 mod |O_{k_{s}}|). Also, if N_{n} > 2 max_{1≤i≤s} |O_{k_{i}}| and h_{n}(x^{1}, …, x^{r+1})_{0} ≠ 0 and h_{n}(x^{1}, …, x^{r+1})_{1} = 0, then h_{n}(x^{1}, …, x^{r+1})_{m} = 0 for 1 ≤ m ≤ N_{n}/2. Thus if N_{n} is large enough we get that h_{n}(S^{r+1}) has entropy < 2^{−n} · ε.

Now fix a sequence N_{n} such that for each n it holds that

h(h_{n}(S^{r+1})) < 2^{−n} · ε.

We have T_{1} = S and thus h(T_{1}) = log2. We now estimate the entropy of T_{n+1} for n ≥ 1. For every 1 ≤ m ≤ n let 1 ≤ s(m) ≤ r denote the integer such that i(m) ∈ ℕ^{s(m)}. Fix j ∈ ℕ. Now we consider the map (h_{1}, …, h_{n}): S^{r+1} → h_{1}(S^{r+1}) × ⋯ × h_{n}(S^{r+1}). Fix (a^{1}, …, a^{n}) ∈ B_{j}((h_{1}, …, h_{n})(S^{r+1})). Let A = A(a^{1}, …, a^{n}) := {(x^{1}, …, x^{r+1})|h_{m}(x^{1}, …, x^{r+1})[0, j) = a^{m} for 1 ≤ m ≤ n}. Let 0 ≤ i < j and let (x^{1}, …, x^{r+1}) ∈ A.

If a_{i}^{m} = 0 for all 1 ≤ m ≤ n then f_{m}(x^{1}, …, x^{r+1})_{i} = 0 for all 1 ≤ m ≤ n and f_{0}(x^{1}, …, x^{r+1})_{i} ∈ {0, 1}. Thus #{(f_{0}, …, f_{n})(x^{1}, …, x^{r+1})_{i}|(x^{1}, …, x^{r+1}) ∈ A(a^{1}, …, a^{n})} ≤ 2.

Now assume there is 1 ≤ m ≤ n with a_{i}^{m} ≠ 0. Consider all 1 ≤ k ≤ n with a_{i}^{k} ≠ 0. Choose k among those with maximal s(k). Then the definition of h_{k} implies that x_{i}^{1}, …, x_{i}^{s(k)} are determined by a_{i}^{k}. Thus f_{0}(x^{1}, …, x^{r+1})_{i} = x_{i}^{1} is determined by a_{i}^{k} and for 1 ≤ m ≤ n we get that in case that (s(m) < s(k) or a_{i}^{m} = 0) that f_{m}(x^{1}, …, x^{r+1})_{i} is uniquely determined by a_{i}^{m} and a_{i}^{k} and in case that (s(m) = s(k) and a_{i}^{m} ≠ 0) we get that f_{m}(x^{1}, …, x^{r+1})_{i} = x_{i}^{s(k)+1}. Again #{(f_{0}, …, f_{n})(x^{1}, …, x^{r+1})_{i}|(x^{1}, …, x^{r+1}) ∈ A} ≤ 2.

Thus for each (a^{1}, …, a^{n}) ∈ B_{j}((h_{1}, …, h_{n})(S^{r+1})) it holds that

#{(f_{0}, …, f_{n})(x^{1}, …, x^{r+1})[0, j)|(x^{1}, …, x^{r+1}) ∈ A(a^{1}, …, a^{n})} ≤ 2^{j}.

Thus

B_{j}(T_{n+1}) = #{(f_{0}, …, f_{n})(x^{1}, …, x^{r+1})[0, j)|(x^{1}, …, x^{r+1}) ∈ S^{r+1}}
≤ 2^{j} · #B_{j}((h_{1}, …, h_{n})(S^{r+1})).

Thus h(T_{n+1}) ≤ log2 + h((h_{1}, …, h_{n})(S^{r+1})) < log2 + ε.

We now estimate the residual entropy of T. For that we use the following two general lemmata.

Lemma D.6. Let R be a subshift and g: R → S be a quotient map and r ∈ ℕ such that

h(g^{−1}(O)) ≥ log(2^{r}) for each finite orbit O of S.

Then h(R) ≥ log(2^{r+1}).

Proof of Lemma D.6. Let m be a coding length for g, that is g(x)_{i} is determined by x[−m+i, i+m] for all x ∈ R and all i ∈ ℤ. Let b ∈ B_{n}(S) and let O(b) denote the orbit of b^{∞}. Let A(b) := {x[−m, n+m)|gx[0, n) = y[0, n) for some y ∈ O(b)}. Since h(g^{−1}(O(b))) ≥ log2^{r} we get #A(b) ≥ 2^{r(2m+n)}. Since O(b) ≠ O(b′) implies that A(b) is disjoint from A(b′) and the map b → O(b) is at most n−to−1

we get #B_{2m+n}(R) ≥ ∑_{b∈B_{n}(S)} n^{−1} ·#A(b) ≥ n^{−1} ·2^{r(2m+n)} ·2^{n} = n^{−1} ·2^{n(r+1)+2rm}.

This holds for all n and thus h(R) ≥ log(2^{r+1}). □

Lemma D.7. Let R be a subshift and g : R → S be a quotient map. Let s ∈ ℕ. Assume there is a family of quotient maps g_{k}: R → S, k ∈ ℕ ∪ ℕ^{2} ∪ ⋯ ∪ ℕ^{s} such that for each (j_{1}, …, j_{s+1}) ∈ ℕ^{s+1} it holds that

h(g^{−1}O_{j_{1}} ∩ g_{j_{1}}^{−1}O_{j_{2}} ∩ g_{(j_{1},j_{2})}^{−1}O_{j_{3}} ∩ ⋯ ∩ g_{(j_{1},...,j_{s})}^{−1}O_{j_{s+1}}) ≥ log2.

Then h(R) ≥ log(2^{s+2}).

Proof of Lemma D.7. Let

R(j_{1}, …, j_{s}) := g^{−1}O_{j_{1}} ∩ g_{j_{1}}^{−1}O_{j_{2}} ∩ g_{(j_{1},j_{2})}^{−1}O_{j_{2}} ∩ ⋯ ∩ g_{(j_{1},...,j_{s−1})}^{−1}O_{j_{s}}.

Then R(j_{1}, …, j_{s}) is a subshift and α_{1} := g_{(j_{1},...,j_{s})}|_{R(j_{1},...,j_{s})} is a quotient map onto S with h(α_{1}^{−1}(O)) ≥ log2 for all finite orbits O of S. Thus Lemma D.6 with r = 1 implies h(R(j_{1}, …, j_{s})) ≥ log4. Thus R(j_{1}, …, j_{s−1}) is a subshift and α_{2} := g_{(j_{1},...,j_{s−1})}|_{R(j_{1},...,j_{s−1})} is a quotient map onto S with h(α_{2}^{−1}(O)) ≥ log4 for all finite orbits O of S. Thus Lemma D.6 with r = 2 implies h(R(j_{1}, …, j_{s−1})) ≥ log8. Inductively one obtains in this way h(R(j_{1})) ≥ log(2^{s+1}) for all j_{1} and thus a final application of Lemma D.6 with r = s + 1 shows h(R) ≥ log(2^{s+2}). □

Now let R be a subshift and φ: R → T be a quotient map. Let φ_{n}: R → T_{n} denote the map φ followed by the projection from T onto T_{n}. For n ≥ 1 let pr_{n}: T_{n} → S denote the projection onto the last coordinate. Let π_{i,j}: T_{j+1} → T_{i} denote the composition of the maps π_{j}, …, π_{i}.

If r = 1 then by Lemma D.6 it suffices to show that h(φ_{1}^{−1}O) ≥ log2 for every finite orbit O of S. Since r = 1 we have that the map i: ℕ → ℕ is the identity. Let n ≥ 1. Then pr_{n+1}((f_{0}, …, f_{n})(O_{n} × S)) = f_{n}(O_{n} × S) = S by definition of f_{n}. Thus h(φ_{1}^{−1}O_{n}) = h((π_{1,n}φ_{n+1})^{−1}O_{n}) ≥ h((π_{1,n})^{−1}O_{n}) = h((f_{0}, …, f_{n})(O_{n} × S)) ≥ log2.

Now consider the case that r > 1. We shall apply Lemma D.7. We define g := φ_{1} : R → S. For k = (j_{1}, …, j_{s}) ∈ ℕ ∪ ℕ^{2} ∪ ⋯ ∪ ℕ^{r−1} let n = n(k) such that i(n − 1) = (j_{1}, …, j_{s}). Then define g_{k} := pr_{n(k)}φ_{n(k)} : R → S. Now let (j_{1}, …, j_{r}) ∈ ℕ^{r}. Let R(j_{1}, …, j_{r}) := g^{−1}O_{j_{1}} ∩ (g_{j_{1}})^{−1}O_{j_{2}} ∩ (g_{(j_{1},j_{2})})^{−1}O_{j_{3}} ∩ ⋯ ∩ (g_{(j_{1},...,j_{r−1})})^{−1}O_{j_{r}}. Choose n_{m} such that i(n_{m} −1) = (j_{1}, …, j_{m}) for 1 ≤ m ≤ r and let n_{0} = 0. (Here n_{m} ≤ n_{r} by definition of ≺ and the bijection i.) Let P := O_{j_{1}} ×⋯×O_{j_{r}} ×S. Let x ∈ R such that φ_{n_{r}}(x) ∈ (f_{0}, …, f_{n_{r}−1})(P) ⊂ T_{n_{r}}. Then g(x) = φ_{1}(x) = π_{1,n_{r}}φ_{n_{r}}(x) ∈ f_{0}(O_{j_{1}}) = O_{j_{1}}. Thus x ∈ g^{−1}O_{j_{1}}. Now let 1 ≤ m < r. We show x ∈ (g_{(j_{1},...,j_{m})})^{−1}O_{j_{m+1}}. We have g_{(j_{1},...,j_{m})}(x) = pr_{n_{m}}φ_{n_{m}}(x). Since φ_{n_{m}}(x) = π_{n_{m},n_{r}}φ_{n_{r}}(x) ∈ (f_{0}, …, f_{n_{m}−1})(P) we get g_{(j_{1},...,j_{m})}(x) ∈ f_{n_{m}−1}(P) ⊂ O_{j_{m+1}}. Thus (f_{0}, …, f_{n_{r}−1})(P) ⊂ φ_{n_{r}}(R(j_{1}, …, j_{r})) and since pr_{n_{r}}((f_{0}, …, f_{n_{r}−1})(P)) = f_{n_{r}−1}(P) = S by definition of f_{n_{r}−1}, it follows that h(R(j_{1}, …, j_{r})) ≥ h((f_{0}, …, f_{n_{r}−1})(P)) ≥ log2. Thus the assumptions of Lemma D.7 are satisfied with s = r−1 and thus we have that h(R) ≥ log(2^{r+1}). This completes the proof of Proposition D.5, and therefore completes the proof of Theorem D.1.

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Department of Mathematics, University of Maryland, College Park, MD 20742-4015, U.S.A.

E-mail address: mmb@math.umd.edu

Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, 37083 Göttingen, Germany

E-mail address: fiebig@math.uni-goettingen.de

Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany

E-mail address: fiebig@math.uni-heidelberg.de