THE ENTROPY THEORY OF SYMBOLIC EXTENSIONS

MIKE BOYLE AND TOMASZ DOWNAROWICZ

Abstract. Fix a topological system (X, T), with its space K(X, T) of T - invariant Borel probabilities. If (Y, S) is a symbolic system (subshift) and φ : (Y, S) → (X, T) is a topological extension (factor map), then the function h_{ext}^{φ} on K(X, T) which assigns to each μ the maximal entropy of a measure ν on Y mapping to μ is called the extension entropy function of φ. The infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by h_{sex}. In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X, T). The entropy structure H is a sequence of entropy functions h_{k} defined with respect to a refining sequence of partitions of X (or of X × Z, for some auxiliary system (Z, R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities. We develop the functional analysis and computational techniques to produce many dynamical examples; for instance, we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of (X, T) must always be attained, and we show that the maximum value of h_{sex} need not be achieved at an ergodic measure. We exhibit several characterizations of the asymptotically h-expansive systems of Misiurewicz, which emerge as a fundamental natural class in the context of the entropy structure. The results of this paper are required for the Downarowicz-Newhouse results [DN] on smooth dynamical systems.

Contents

1. Introduction 2 2. Semicontinuity, separation theorems and superenvelopes 4 3. Estimates and inductive characterization of EF 9 4. Affine functions and attainability 13 5. The Sex Entropy Theorem 16 6. Proof of the Sex Entropy Theorem: a zero dimensional case 17 7. Proof of the Sex Entropy Theorem: the general case 27 8. Consequences for sex entropy 31 References 35

Date: July 31, 2003. 2000 Mathematics Subject Classification. Primary: 37B10; Secondary: 37B40, 37C40, 37C45,

37C99, 37D35.

Key words and phrases. sex entropy, residual entropy, conditional entropy, topological entropy, entropy, superenvelope, symbolic dynamics.

The research of the second named author was supported by a KBN grant.

1. Introduction

Suppose S, T are selfhomeomorphisms of compact metrizable spaces X, Y and φ: Y → X is a continuous surjection such that T φ = φS. Then we refer to φ, and also (Y, S), as an extension of (X, T). We say it is a symbolic extension if (Y, S) is a subshift over a finite alphabet. Symbolic extensions are fundamental to the use of symbolic dynamics in the study of dynamical systems (e.g. [Bow]).

We introduce h_{sex}(T), the symbolic extension entropy of T, by setting

h_{sex}(T) = inf{h_{top}(S): (Y, S) is a symbolic extension of (X, T)}

where h_{top}(S) denotes the topological entropy of S. We adopt the convention that the infimum of the empty set is by definition ∞. Abbreviating, we use “sex entropy” as synonymous with “symbolic extension entropy”. Sex entropy was studied [BFF, D2] in the notation of the residual entropy of T, which is by definition h_{res}(T) = h_{sex}(T) − h_{top}(T). The residual entropy gives a coarse measure of describability of a system by symbolic dynamics. The notational choice of sex entropy will be more natural for most of the current paper (e.g., Remark 8.2).

We define the extension entropy function of an extension φ: (Y, S) → (X, T) as

h_{ext}^{φ}: K(X, T) → [0, ∞)
μ ↦ sup{h(S, ν): ν ∈ K(Y, S) and φν = μ}

where e.g. K(X, T) denotes the compact space of T -invariant Borel probabilities, h(S, ν) is the measure theoretic entropy of S with respect to the measure ν, and we let φ abbreviate the induced map φ_{∗} on measures. For a given μ, the number h_{ext}^{φ}(μ) measures the complexity of the orbits appearing in Y above the support of μ. There is a variational principle ([LeWa, DS1]) which gives a precise a.e. formulation of this idea in terms of fiber entropies (see 6.5).

In this paper, we will study entropy jumps to symbolic systems at the level of measures. We introduce h_{sex}(T, ·), the symbolic extension entropy function of T, by setting

h_{sex}(T, μ) = inf{h_{ext}^{φ}(μ): φ is a symbolic extension of T }

and similarly we define the residual entropy function of T on K(X, T) by h_{res}(T, μ) = h_{sex}(T, μ) − h(T, μ). Sex entropy as a function on measures provides a much finer probe into the dynamics of T than the number h_{sex}(T). We will obtain a rather complete functional analytic understanding of sex entropy functions.

For this, given a system (X, T), we define an entropy structure, generalizing to arbitrary systems the entropy structure introduced in [D2] in dimension zero as a key tool for studying residual entropy. First, suppose the system (X, T) admits a refining sequence of finite Borel partitions P_{k} with small boundaries (i.e., boundaries which have measure zero for every measure in K(X, T)). In this case (a common one – see 7.6), we may define an entropy structure for (X, T) to be the sequence H of functions h_{k}: K(X, T) → ℝ defined by the rule h_{k}: μ ↦ h(T, μ, P_{k}). For a general system (X, T), we define the sequence H with respect to (X × Z, T × R), where (Z, R) can be any nonperiodic zero entropy minimal system (see 5.2). The point of this is that (X × Z, T × R) admits the required refining sequence P_{k} with small boundaries, as a consequence of the deep constructions of Lindenstrauss [Li], using the mean dimension theory of dynamical systems developed by Lindenstrauss and Weiss [LiWe]. Our results will not depend on the particular sequence H chosen for a system (there is no canonical sequence). In any case H will be a nondecreasing sequence of affine nonnegative upper semicontinuous (u.s.c.) functions with u.s.c. differences.

We define a superenvelope of the sequence H to be a function E such that for each k the difference E −h_{k} is nonnegative and u.s.c. We allow just one unbounded superenvelope, E ≡ ∞. Our key result is the Sex Entropy Theorem (5.5): If H is an entropy structure for (X, T), then E is a bounded affine superenvelope for the entropy sequence H if and only if there exists a symbolic extension φ of (X, T) such that E = h_{ext}^{φ}.

Consequently, we identify h_{sex}(T, ·) as the infimum of all affine superenvelopes of H, which equals EH, the smallest superenvelope of the entropy structure. We give two additional characterizations of this function, along with practical methods for estimation. The Entropy Structure Realization Theorem of [DS2] (see 8.4) shows that every nondecreasing bounded sequence of affine u.s.c. functions with u.s.c. differences on a Choquet simplex is realized as the entropy structure of a dynamical system. Along with the Sex Entropy Theorem, this reduces various questions of sex entropy to problems (or exercises) in functional analysis.

From here we describe the organization of the paper.

The work in the sections 2-4 is pure functional analysis. We use F to denote a nondecreasing sequence of nonnegative u.s.c. functions f_{k} with u.s.c. differences f_{k+1} − f_{k} and lim f_{k} = f. Entropy results follow by replacing F with H. In Section 2, we introduce functional analytic background necessary for the sequel, and we characterize superenvelopes in terms of “continuous covers” of F. In Section 3, we study the smallest superenvelope EF of F. We give an inductive characterization of EF in terms of tail defects of upper semicontinuity along F, using a construction which is often practical but in general can be unavoidably transfinite. We also provide estimates for EF. Work from this section is used in [DN], which provides the first C^{r} (1 ≤ r < ∞) examples of positive residual entropy. (If T is C^{∞}, then h_{res}(T, ·) ≡ 0 [BFF].) In Section 4, we give material on affine functions necessary to later entropy attainability results.

In Section 5, we give a careful statement of our key result, the Sex Entropy Theorem. In Section 6, we prove a zero dimensional case (the main argument) of this theorem. We explain the passage to the general case in Section 7. In Section 8, we assemble and discuss the applications of the preceding sections to symbolic extensions, including a variational principle for sex entropy and several attainability (and nonattainability) results. For the sex entropy and residual entropy functions, maximum values need not be attained on ergodic measures – a rare situation in ergodic theory. If h_{sex}(T, ·) is affine (which is always the case when K(X, T) is a Bauer simplex), then it will be the extension entropy function of a symbolic extension of T. On the other hand, it can happen that no symbolic extension of T has topological entropy equal to h_{sex}(T). We exhibit additional characterizations of asymptotic h-expansiveness, which is a very natural condition in the entropy structure/sex entropy context: T is asymptotically h-expansive if and only if it has a principal extension to a symbolic system (i.e., the sex entropy function h_{sex} equals h), if and only if h_{k} converges to h uniformly. We explain how the characterization in [D2] of the residual entropy of a system is a consequence of the superenvelope results. By mixing functional analytic examples of the earlier sections with the Entropy Structure Realization Theorem, we produce several examples of residual entropy phenomena. To be concrete, we end the section with two explicit constructions of dynamical systems: one with positive residual entropy, and another with zero residual entropy but with no symbolic extension of equal entropy.

Our entropy structures use refining sequences of partitions with small boundaries. To avoid partition constructions and the invocation of an auxiliary system in our general construction of an entropy structure (5.2(II)), and to get a better understanding of the dynamical meaning of sex entropy, it is appropriate to examine the relation of entropy to scale in terms of shrinking covers, continuous functions, epsilon-orbits with epsilon going to zero, and so on. Not all natural definitions give an entropy structure whose bounded affine superenvelopes are the sex entropy functions of symbolic extensions. The appropriate structures and equivalence relations are explained in [D3].

We thank the referees for their careful readings of the paper, and for their numerous useful suggestions which have improved it.

2. Semicontinuity, separation theorems and superenvelopes

This section contains certain general observations concerning the behavior of sequences of u.s.c. functions on compact metrizable spaces, including, in particular, affine u.s.c. functions on convex sets (simplexes). (For our applications we only consider the metrizable case, although most statements below hold without the metrizability assumption.)

Upper semicontinuous functions We begin by recalling some basic facts about u.s.c. functions.

Definition 2.1. A function f: K ↦ ℝ defined on a topological Hausdorff space K is called upper semicontinuous (u.s.c.) if one of the following equivalent conditions holds.

(1) f = inf_{α} g_{α} for some family {g_{α}} of continuous functions.

(2) f = lim_{α} g_{α}, where (g_{α}) is a nonincreasing net (sequence in metric spaces) of continuous functions.

(3) For each r ∈ ℝ, the set {x: f(x) ≥ r} is closed.

(4) lim sup_{x′→x} f(x′) ≤ f(x) at each x ∈ K.

Properties 2.2. (1) The infimum of any family of u.s.c. functions is again

u.s.c. (by 2.1(1)).

(2) Both the sum and the supremum of finitely many u.s.c. functions are u.s.c. (by 2.1(3)).

(3) A uniform limit of u.s.c. functions is u.s.c.

(4) If f = lim f_{α} is the limit of a nonincreasing net of u.s.c. functions defined on a compact domain K, and g > f is a continuous function, then g > f_{α} for some α. In particular, a nonincreasing convergence of u.s.c. functions to a continuous limit is always uniform.

(5) Every u.s.c. function from a compact domain to ℝ is bounded above and attains its maximum.

Notation 2.3. SUP(f) will be used to denote the supremum of f, i.e. sup_{x∈K} f(x), while sup_{α} f_{α} always means the function defined by pointwise supremum. We use INF and inf similarly.

Proposition 2.4 (Exchanging Suprema and Infima). If (f_{α}) is a nonincreasing net of u.s.c. functions on a compact space, then inf_{α} SUP(f_{α}) = SUP(inf_{α} f_{α}).

Proof. The inequality “≥” is obvious. To prove the converse, suppose that

inf_{α} SUP(f_{α}) > SUP(inf_{α} f_{α}) .

Viewing the left hand side as a constant function g we have g > inf_{α} f_{α}. Thus, by Property 2.2(4), g > f_{α}(x) for some α and every point x, including the point where f achieves SUP. But g ≤ SUP(f_{α}) by definition, a contradiction. □

We now pass to lifting and projecting u.s.c. functions. Let π: M → K be a continuous map. If f is a continuous (or u.s.c.) function on K then the lifted function f ∘ π is a continuous (or u.s.c.) function on M. This lift we denote by the same letter f. We can also push functions down:

Definition 2.5. If π: M → K is a surjection and f is a bounded real-valued function on M, then f^{[K]} is the function defined on K by

f^{[K]}(x) = sup_{y∈π^{−1}(x)} f(y) .

Remark 2.6. In the case that f is u.s.c. and π is a continuous surjection between compact spaces, it is easily seen using Definition 2.1(3) that f^{[K]} is also u.s.c.

Below, exK denotes the set of extreme points of a convex set K. A Choquet simplex is a compact convex subset K of a locally convex space such that the dual of the continuous affine functions on K is a lattice [AE, p.69]. We will only need the characterization of Choquet: a metrizable compact convex subset K of a locally convex space is a Choquet simplex if and only if for each x in K there exists a unique Borel probability μ_{x} supported on exK such that ∫ fdμ = f(x) for every continuous affine function on K [P1, p.60]. Every Choquet simplex is affinely homeomorphic to K(X, T) for some homeomorphism T of a Cantor set [D1, O]; here the integral representation is the ergodic decomposition of a measure.

Proposition 2.7. If π: M → K is a continuous affine surjection between convex sets, and f is an affine function defined on M, then f^{[K]} is concave. If in addition M and K are Choquet simplexes and π preserves the extreme points (i.e., π(exM) ⊂ exK), then f^{[K]} is affine.

Proof. Let x = αx_{1} + βx_{2} in K (α ∈ (0, 1), β = 1 − α). Clearly

π^{−1}(x) ⊃ {αy_{1} + βy_{2} : y_{1} ∈ π^{−1}(x_{1}), y_{2} ∈ π^{−1}(x_{2})}

hence concavity follows. For the reversed inclusion, and hence convexity of f^{[K]} (with all the additional assumptions) we will apply the Choquet Representation Theorem and the Radon-Nikodym Theorem. Let μ_{i} be the unique probability measure supported by exK with barycenter at x_{i}, (i = 1, 2). Clearly, μ = αμ_{1}+βμ_{2} is the unique probability measure supported by exK with barycenter at x, and each μ_{i} is absolutely continuous with respect to μ. Let f_{i} denote the corresponding Radon-Nikodym derivative defined on exK. Note that αf_{1} + βf_{2} ≡ 1. Let ν be the probability measure on exM with barycenter at a chosen point y ∈ π^{−1}(x). Consider the measures ν_{i} on exM defined by dν_{i} = (f_{i} ∘ π)dν (here we use the assumption π(exM) ⊂ exK). We have

ν_{i}(exM) = ⟨f_{i} ∘ π, ν⟩ = ⟨f_{i}, π(ν)⟩ .

Since the barycenter of π(ν) coincides with the image by π of the barycenter of ν, i.e., with π(y) = x, by the uniqueness of such measure, we have π(ν) = μ. Thus

ν_{i}(exM) = ⟨f_{i}, μ⟩ = 1,

which shows that each ν_{i} is a probability measure. Then we let y_{i} be the barycenter of ν_{i}. The image π(y_{i}) coincides with the barycenter of the measure π(ν_{i}), and it is straightforward to verify that π(ν_{i}) = μ_{i}, hence π(y_{i}) = x_{i} (i = 1, 2). Finally,

αdν_{1} + βdν_{2} = ((αf_{1} + βf_{2}) ∘ π)dν = dν,

hence, passing to barycenters, αy_{1} + βy_{2} = y, which completes the proof. □

Separation theorems

A function f is called lower semicontinuous (l.s.c.) if −f is u.s.c. Essential to the sequel are various results, collected below, on separating l.s.c. and u.s.c. functions. We give some proof details where we lack precise references.

Theorem 2.8 (Sandwich Theorem). Let K be a compact metric space. Suppose h and f are functions defined on a compact metric space K, h ≤ f, h is u.s.c. and f is l.s.c. Then there exists a continuous function g such that h ≤ g ≤ f.

Proof. See e.g. [T]. (The result holds for a topological space K if and only if K is normal.) □

Theorem 2.9 (Separation of Disjoint Epigraphs). Suppose h and f are functions defined on a compact convex subset K of a locally convex linear space, h < f, h is u.s.c. and f is l.s.c.

(1) If h is concave and f is convex, then there exists an affine continuous function g such that h < g < f.

(2) If h is concave, then there exists a concave continuous function g such that h < g < f.

Proof. (1) This is [C, Theorem 21.20], which we include for context.

(2) The concave u.s.c. function h is the pointwise infimum of the continuous affine functions a such that h < a (see [C, Prop. 2.18]). Because f is l.s.c. with h < f, it follows that for each x ∈ K there is an open neighborhood V and an affine function g_{V} such that h < g_{V} and g_{V} (y) < f(y) for y ∈ V. Let g = min g_{V}, where the minimum is over a finite cover of K by such sets V. □

Theorem 2.10 (Edwards’ Separation Theorem). Suppose h and f are functions defined on a Choquet simplex K, h ≤ f, h is convex and f is concave l.s.c. Then there exists an affine continuous function g such that h ≤ g ≤ f.

Proof. See e.g. [AE], Theorem 7.6. □

Theorem 2.11. Suppose h and f are functions defined on a Choquet simplex, h < f, h is affine u.s.c. and f is l.s.c. Then there exists an affine continuous function g such that h < g < f.

Proof. Choose ϵ > 0 such that h+ϵ < f. By Theorem 2.9(2), there exists a concave continuous function g_{1} such that h + ϵ < g_{1} < f. By Theorem 2.10, there exists an affine continuous function g_{2} such that h + ϵ ≤ g_{2} ≤ g_{1}. Let g = g_{2}. □

Envelopes and superenvelopes

Let f be a bounded function defined on a compact domain K. We assume the set K is also convex in discussions involving convex structure. Throughout this paper we will be using the following notation:

Notation 2.12. f̃ denotes the u.s.c. envelope of f, i.e., f̃ = inf_{α} g_{α}, where g_{α} ranges over all continuous functions above f (i.e., g_{α} ≥ f).

In fact, f̃ is the smallest u.s.c. function above f, and

f̃(x) = max {f(x), lim sup_{x′→x} f(x′)} .

It is also immediately seen that for any functions f and g, { \atop f + g} ≤ f̃ + g̃, with equality holding if f or g is continuous.

Definition 2.13. f̂ denotes the u.s.c. concave envelope of f, i.e., f̂ = inf_{α} g_{α}, where g_{α} range over all affine continuous functions above f.

Remarks 2.14. Clearly, f ≤ f̃ ≤ f̂. In fact, f̂ is the smallest concave u.s.c. function above f (see [C, Prop. 2.18]). If f is affine then f̃ is concave, hence f̃ = f̂. Indeed, let x = αx_{1} + βx_{2}, and (using the above displayed formula for f̃) choose sequences x_{i,n} → x_{i}, (i = 1, 2) such that f(x_{i,n}) → f̃(x_{i}). Then

f̃(x) ≥ lim_{n} f(αx_{1,n} + βx_{2,n}) = lim_{n}[αf(x_{1,n}) + βf(x_{2,n})] = αf̃(x_{1}) + βf̃(x_{2}).

Notation 2.15. If f is unbounded, we set f̃ = f̂ ≡ ∞ which is to say that these envelopes do not exist (in the proper sense). Obviously, in any case SUP(f) = SUP(f̃) = SUP(f̂).

For the rest of this section, we set the following

Notation 2.16. F denotes (f_{k})_{k=1}^{∞}, a nondecreasing sequence of u.s.c. functions f_{k}: K → [0, +∞), where K is compact metrizable, all differences f_{k+1} − f_{k} are u.s.c., and lim_{k} f_{k} = f. We allow ∞ as a possible value of f. We let f_{0} denote the zero function.

Definition 2.17. A superenvelope of F is a function E on K such that either (i) E is bounded and for each k ≥ 1 the function E − f_{k} is nonnegative and u.s.c., or (ii) E is the constant function ∞.

Suppose E is a finite superenvelope of F. Then E = (E − f_{1}) + f_{1}, so E is itself a u.s.c. function. The functions E − f_{k} converge nonincreasingly to the function E − f, which hence is also u.s.c. and nonnegative. Since the pointwise minimum of two superenvelopes is easily seen to be again a superenvelope, the superenvelopes of F form a directed family. It follows immediately from Property 2.2(1) that there exists a smallest superenvelope of F, namely, the pointwise infimum (limit) of all superenvelopes. We denote that superenvelope by EF. This function is either bounded or it is the constant ∞.

We will now provide our first construction of the smallest superenvelope EF. By a continuous cover (or simply cover) of F we shall mean any sequence G = (g_{k})_{k∈ℕ} of continuous functions from K to [0, ∞] such that g_{k} ≥ f_{k} − f_{k−1} for each k ∈ ℕ. Let ΣG denote Σ_{k=1}^{∞}g_{k}. We will consider the u.s.c. function

inf_{G} Σ̃G,

where the infimum is taken over all covers G of F.

Theorem 2.18. EF = inf_{G} Σ̃G. There exists a cover G with SUP(ΣG) = SUP(EF).

Proof. Notice that the set {G} of all covers of F is a directed family, with the order G ≽ G′ if g_{k} ≤ g′_{k} for each k; the common successor of two covers is obtained as their pointwise minimum. By attaching to each cover G the function Σ̃G, we regard {Σ̃G} as a net indexed by G. This is a nonincreasing net of u.s.c. functions, and the infimum in the definition of inf_{G} Σ̃G is in fact a nonincreasing limit.

Let F_{k} denote the sequence (f_{n+k} − f_{k})_{n∈ℕ}. We can write

inf_{G} Σ̃G = lim_{G} (Σ_{i=1}^{k}g_{i}+Σ̃G_{k}) = f_{k} + inf_{G} Σ̃G_{k} ,

where G_{k} = (g_{n+k})_{n∈ℕ} is the cover of F_{k} obtained by “truncating” G (in fact all covers of F_{k} are such truncations). The difference between the function inf_{G} Σ̃G and f_{k} is hence a nonnegative u.s.c. function. We have proved that the function inf_{G} Σ̃G is a superenvelope of F, hence it is not smaller than EF.

The converse inequality holds trivially if EF ≡ ∞. Suppose EF is bounded and let g ≥ EF be a continuous function. We will inductively construct a cover G such that ΣG ≤ g. This will end the proof, because then also Σ̃G ≤ g and hence

inf_{G} Σ̃G ≤ inf_{g≥EF} g = EF.

(1) It is clear that

f_{1} ≤ g − (EF − f_{1})

and in this inequality the assumptions of the Sandwich Theorem 2.8 hold. Thus, there exists a continuous function g_{1} in between.

(2) Suppose we have found continuous functions g_{1}, g_{2}, …, g_{k} such that we have g_{i} ≥ f_{i} − f_{i−1} for i = 1, 2, …, k, and

∑_{i=1}^{k} g_{i} ≤ g − (EF − f_{k}) .

(This condition was fulfilled in (1) above for the case k = 1.) Then

f_{k+1} − f_{k} ≤ g − ∑_{i=1}^{k} g_{i} − (EF − f_{k+1})

and again, the assumptions of the Sandwich Theorem are being fulfilled. Thus a continuous function g_{k+1} exists between the expressions on both sides. It is seen that the inductive assumption is now satisfied for k + 1. In this manner a cover G = (g_{k})_{k∈ℕ} of F is constructed. Finally, since each EF − f_{k} is nonnegative, Σ_{i=1}^{k}g_{i} ≤ g holds for each k, hence ΣG ≤ g.

The cover mentioned in the last statement of the theorem is obtained by applying the last argument to the constant function g ≡ SUP(EF). □

Example 2.19. Let K consist of a sequence (a_{n})_{n≥2} and its limit b. Define f_{1} = 1_{{b}} and f_{k} = f_{k−1} + 1_{{a_{k}}} if k ≥ 2. Clearly f ≡ 1, and it is not hard to see using continuous covers (or Proposition 3.13) that EF = 1_{{b}} + 1. In this example, it is seen that f being u.s.c. (even continuous) does not imply that it is equal to EF.

Example 2.20. Let K be a space without isolated points, and let (q_{n})_{n∈ℕ} be a dense sequence. Set f_{1} = 1_{q_{1}} and f_{k} = f_{k−1} + 1_{{q_{k}}} if k ≥ 2. Clearly f ≤ 1; however, EF ≡ ∞. To see this, check that ΣG is unbounded for any continuous cover G, or apply Proposition 3.9.

Example 2.21. Let K be the following subset of ℝ^{2}:

K = \{(0, 0)\} ∪ \{(\frac{1}{m}, 0): m ∈ ℕ\} ∪ \{(\frac{1}{m}, \frac{1}{n}): (m, n) ∈ ℕ^{2}, m ≤ n\} .

For k ∈ ℕ and given a, b, c ∈ ℝ, define f_{k} = ∑_{i=1}^{k} d_{i}, where

d_{1} = a · 1_{\{(0,0)\}} ,  d_{2k} = b · 1_{\{( \frac{1}{k} ,0)\}} ,  d_{2k+1} = c · 1_{\{( \frac{1}{m_{k}} , \frac{1}{n_{k}} )\}} ,

and (m_{k}, n_{k}) is some linear ordering of the set {(m, n) ∈ ℕ^{2} : m ≤ n}. Clearly SUP(f) = max{a, b, c}. By analyzing continuous covers (or computing EF from Theorem 3.3), one can check EF ((\frac{1}{m}, \frac{1}{n})) = c, EF ((\frac{1}{m}, 0)) = b+c, and EF ((0, 0)) = a + b + c. Playing with higher order accumulation points in K, we can obtain arbitrary real numbers 0 < r_{1} ≤ r_{2} as SUP(f) and SUP(EF).

3. Estimates and inductive characterization of EF

Throughout this section we continue the notation and assumptions of (2.16), and we also assume that f is bounded (otherwise there is no point in estimating EF). First of all, it is important to realize that when EF is bounded, the functions EF and f must be equal on a large set.

Proposition 3.1. If EF < ∞, then EF = f on a residual subset of K.

Proof. Suppose EF − f ≥ ϵ on a set containing an open set U. Then EF − ϵ1_{U} is easily seen to be a superenvelope of F, which contradicts minimality of EF. We have shown that the set where EF − f ≥ ϵ has empty interior. This set is also closed, so EF > f on a first category set. □

On the other hand, for f to equal EF exactly is rather special.

Proposition 3.2. The following conditions are equivalent:

(1) EF = f

(2) f_{k} converges to f uniformly on K.

Proof. If (1) holds, then f − f_{k} is a sequence of u.s.c. functions which converge nonincreasingly to zero. By Property 2.2(4), the convergence is uniform, so (2) holds.

If (2) holds, then every f −f_{k} is a uniform limit of u.s.c. functions, hence is u.s.c., so f is a superenvelope, and obviously it is the minimal one, so (1) holds. □

Evidently the mystery of EF −f has to do with upper semicontinuity and nonuniform convergence of the sequence F. We will capture this with an inductive characterization, which is often practical but in general transfinite. Given a nonincreasing sequence T = (t_{k}) of nonnegative bounded functions on K, we set u_{0} ≡ 0, and then for an ordinal α set

u_{α+1} = lim_{k→∞} { \atop u_{α} + t_{k}} .

Notice that if u_{α} is bounded, then { \atop u_{α} + t_{k}} converges nonincreasingly, and u_{α+1} is also nonnegative u.s.c. and bounded. Finally, for a limit ordinal β let

u_{β} = { \atop sup_{α<β} u_{α}} .

(So, for every α, either u_{α} is bounded or by our convention (2.15) u_{α} ≡ ∞.) We will apply this construction to the tail sequence T = (τ_{k})_{k} of the sequence F, defined by τ_{k} = f − f_{k} for k ≥ 0. For this choice of T we may write u_{α} as u_{α}^{F}.

Theorem 3.3 (Inductive Characterization). Let F be a nondecreasing sequence of nonnegative u.s.c. functions converging to a bounded limit f, and with u.s.c. differences f_{k+1} − f_{k}. Let T = (τ_{k})_{k} denote the sequence of tails. The functions u_{α} = u_{α}^{F} are nondecreasing in α, and

(3.4)  u_{α} = u_{α+1} ⟺ u_{α} = EF − f.

Moreover, such an α exists already among countable ordinals. Consequently,

(3.5)  EF − f = sup_{α} u_{α}

where the supremum is taken over all countable ordinals α, or equivalently over all ordinals α.

Proof. That the functions u_{α} are nondecreasing in α is obvious from the definition. We will first show

(3.6)  u_{α} ≤ EF − f

for any α. This is clear for α = 0, so suppose it holds for some α. Write

EF − f_{k} = EF − f + τ_{k} ≥ u_{α} + τ_{k}.

Because the function on the left is u.s.c., we can replace the function on the right with its u.s.c. envelope, so

EF − f_{k} ≥ { \atop u_{α} + τ_{k}} ≥ u_{α+1}.

Passing with k to infinity we obtain u_{α+1} ≤ EF −f. If β is a limit ordinal and (3.6) holds for α < β, then the inequality u_{β} ≤ EF − f_{k} follows from the definition of u_{β} and the uppersemicontinuity of EF − f. This proves (3.6) for all α.

We will now prove (3.4). If u_{α} ≡ ∞, then by (3.6) the right side of (3.4) holds, and obviously the left side holds. Suppose then u_{α} is bounded. If the right side of (3.4) holds, then

u_{α} + τ_{k} = EF − f + τ_{k} = EF − f_{k}

which is u.s.c., so we also have { \atop u_{α} + τ_{k}} = EF − f_{k}. Thus,

u_{α+1} = lim_{k}(EF − f_{k}) = EF − f = u_{α}.

Now assume the left side of (3.4) holds. It suffices to bound, for each k, the “defect of upper semi-continuity” of u_{α} + τ_{k} as follows:

({ \atop u_{α} + τ_{k}}) − (u_{α} + τ_{k}) ≤ u_{α+1} − u_{α}.

Once this is proved, u_{α+1} = u_{α} implies that u_{α} + τ_{k} is u.s.c. for every k; then u_{α} + τ_{k} = u_{α} + f − f_{k} shows u_{α} + f is a superenvelope; then u_{α} + f ≥ EF, which together with (3.6) gives equality. To estimate these defects first note that they converge to u_{α+1} − u_{α} (because τ_{k} converges to zero). It now suffices to show that they are nondecreasing in k. Let ℓ > k. We have

({ \atop u_{α} + τ_{ℓ}}) − (u_{α} + τ_{ℓ}) − ({ \atop u_{α} + τ_{k}}) + (u_{α} + τ_{k})

= ({ \atop u_{α} + τ_{ℓ}}) − u_{α} − τ_{ℓ} − (u_{α} + τ_{ℓ} + f_{ℓ} − f_{k}) + u_{α} + τ_{ℓ} + f_{ℓ} − f_{k}
= ({ \atop u_{α} + τ_{ℓ}}) + f_{ℓ} − f_{k} − (u_{α} + τ_{ℓ} + f_{ℓ} − f_{k}) ≥ 0

because f_{ℓ} − f_{k} is u.s.c. This completes the proof of the equivalence statement.

For the “Moreover” claim, let K_{α} = {(x, y) ∈ K×[0, ∞]: 0 ≤ y ≤ u_{α}(x)}, a compact set. The space of compact subsets of K × [0, ∞] is separable in the Hausdorff metric, and α < β < γ implies dist(K_{α}, K_{β}) ≤ dist(K_{α}, K_{γ}), by inclusion. Therefore, given ϵ > 0 there can be only countably many α such that dist(K_{α}, K_{α+1}) > ϵ. It follows that u_{α} = u_{α+1} for some countable ordinal α.

The final claim (3.5) is now obvious. □

Examples (which we skip) show that the transfiniteness in the induction is necessary; on the other hand, sometimes one can compute EF by computing u_{α} and u_{α} = u_{α+1} for a finite α (e.g., u_{1} ≠ u_{2} = u_{3} in Example 2.21). The rest of this section will be devoted to some practical estimates for EF. We will use the especially accessible function u := u_{1}^{F} = inf_{k} τ̃_{k} = lim_{k} τ̃_{k}.

Proposition 3.7. Let τ_{k} and u_{1}^{F} be defined as in Theorem 3.3, and set u = u_{1}^{F}. (1) The functions τ̃_{k} − τ_{k} converge nondecreasingly in k to the limit u.

(2) f̃− f ≤ u ≤ f̃.

(3) The pointwise supremum of u is inf_{k} SUP(τ_{k}) = lim_{k} SUP(τ_{k}).

Proof. (1) The proof of Theorem 3.3 showed the defects ({ \atop u_{α} + τ_{k}}) − (u_{α} + τ_{k}) converge nondecreasingly in k to u_{α+1} − u_{α}. The case α = 0 is (1).

(2) The first inequality follows from (1) because f̃− f = τ̃_{0} − τ_{0}, and the second inequality is trivial by the definition of u = u_{1}.

(3) By the Exchanging Suprema and Infima statement of Proposition 2.4,

inf_{k} SUP(τ_{k}) = inf_{k} SUP(τ̃_{k}) = SUP(inf_{k} τ̃_{k}) = SUP(u).
□

We recall a standard definition from topology

Definition 3.8. A point x has order of accumulation 0 in a set K when it is an isolated point in K, i.e., when {x} is open. Let K^{(1)} denote the compact set of all points that are not of order 0. Inductively, x is (an accumulation point) of order r = r(x) (r ≥ 1) in K if it is an isolated point of the compact set K^{(r)} of points which are not of order smaller than r. We set r(x) = ∞ for points which are not of any finite order.

Proposition 3.9. For t > 0, let r(t, x) denote the order of accumulation of x in the set K_{t} of points where u ≥ t. Then EF(x) ≥ f(x) + tr(t, x).

Proof. By the characterization (3.5) of EF − f in Theorem 3.3, it suffices to prove the claim: if r(t, x) ≥ m, then u_{m}(x) ≥ mt. The case m = 0 is trivial. Suppose the claim holds for m, and r(t, x) ≥ m + 1. Choose points x_{n} from K_{t} such that lim x_{n} = x and r(x_{n}) ≥ m. Then u_{m+1}(x) ≥ \overline{lim}_{n} u_{m+1}(x_{n}) = \overline{lim}_{n} lim_{k}(u_{m}(x_{n}) + τ_{k}(x_{n}))^{∼} ≥ \overline{lim}_{n} lim_{k}(mt + τ_{k}(x_{n}))^{∼} ≥ mt + \overline{lim}_{n} u(x_{n}) ≥ (m + 1)t . □

From here we concern ourselves with bounds on EF from above.

Proposition 3.10. Let r(x) denote the order of accumulation of x in the domain K of f. We have the following estimate of the minimal superenvelope EF:

EF(x) ≤ f(x) + r(x)u(x).

Proof. We proceed by induction on r. By Proposition 3.1, EF = f at any isolated point, so the statement holds for r = 0. Suppose we have proved it for some r ≥ 0. Let x be a point of order r + 1. Define E as EF except at x where we set E(x) = f(x) + (r + 1)u(x). We complete the proof by showing that E is a superenvelope of F. The functions E − f_{k} are nonnegative. We only need to verify their upper semi-continuity at x, i.e., given k we need

lim sup_{x′→x}(E − f_{k})(x′) ≤ (E − f_{k})(x).

The point x is surrounded only by points x′ ≠ x of order at most r, for which

(E − f_{k})(x′) − (E − f_{k})(x) = EF(x′) − f_{k}(x′) − f(x) − (r + 1)u(x) + f_{k}(x)
≤ f(x′) + ru(x′) − f_{k}(x′) − f(x) − (r + 1)u(x) + f_{k}(x)
= τ_{k}(x′) − τ_{k}(x) + r(u(x′) − u(x)) − u(x) .

Passing to limsup over x′ → x, we appeal to upper semicontinuity of u and then Proposition 3.7(1) to obtain

lim sup_{x′→x}(E − f_{k})(x′) − (E − f_{k})(x) ≤ τ̃_{k}(x) − τ_{k}(x) − u(x) ≤ 0

as required. □

A refinement of the above method leads to another estimate:

Theorem 3.11. For t > 0 and x ∈ X, define r_{t}(x) to be 0 if u(x) < t, and otherwise define r_{t}(x) to be 1 plus the order of accumulation of x in the set {x: u(x) ≥ t} (i.e., r_{t}(x) = 1 + r(t, x)). Suppose there is an open set V and a function g: ℝ → ℝ such that r_{t}(x) ≤ g(t) for all x ∈ V, and ∫_{0}^{∞} g(t)dt < ∞. Define the nonnegative function

E(x) = f(x) + ∫_{0}^{∞} r_{t}(x)dt .

Then E − f_{k} is u.s.c. on V. If EF is bounded, then EF(x) ≤ E(x) for all x ∈ V.

Proof. The integral is unaffected if the upper limit ∞ is changed to u(x). Let x_{n} be a sequence from V \ {x} with lim x_{n} = x. We need to show for k ∈ ℕ that

lim sup_{n}(E − f_{k})(x_{n}) ≤ (E − f_{k})(x).

Given t with 0 < t ≤ u(x), all points sufficiently close to x have order of accumulation in {x: u(x) ≥ t} smaller than that of x, i.e., lim sup_{n} r_{t}(x_{n}) ≤ r_{t}(x) − 1. For t > u(x), by upper semicontinuity of u, lim sup_{n} r_{t}(x_{n}) = 0. Thus, for all t > 0 we can write lim sup_{n} r_{t}(x_{n}) ≤ max{r_{t}(x) − 1, 0}. By the Dominated Convergence Theorem,

lim sup_{n} ∫_{0}^{∞} r_{t}(x_{n})dt ≤ ∫_{0}^{u(x)} r_{t}(x)dt − u(x)

and therefore (with τ_{k} = f − f_{k})

lim sup_{n}(E − f_{k})(x_{n}) ≤ lim sup_{n} τ_{k}(x_{n}) + lim sup_{n} ∫_{0}^{∞} r_{t}(x_{n})dt
≤ τ̃_{k}(x) + ∫_{0}^{u(x)} r_{t}(x)dt − u(x)
≤ τ̃_{k}(x) + ∫_{0}^{∞} r_{t}(x)dt − (τ̃_{k}(x) − τ_{k}(x))

because u estimates from above the defect of τ_{k}. The last expression, after cancellation, coincides with (E − f_{k})(x).

Now suppose that EF is bounded. If M is a constant such that E ≤ M on V (e.g. M = SUP(f)+∫_{0}^{∞} g(t)dt), then the function which equals E on V and equals EF + M outside V is a superenvelope. This proves the final claim. □

Remark 3.12. The domination assumption cannot be dropped. For instance, in general u(x) = 0 does not imply EF(x) = f(x) (it does whenever Theorem 3.11 applies). For example, x can be the limit of closed sets C_{n} with SUP(u|C_{n}) = \frac{1}{n}, u(x) = f(x) = 0, but SUP(EF|C_{n}) = 1, hence EF(x) = 1.

Proposition 3.13. Suppose j is a nonnegative integer such that f_{k+1} − f_{k} is continuous whenever k ≥ j. Then EF = f_{j} + τ̃_{j} , and SUP(EF) ≤ 2SUP(f). If all f_{k} are continuous, then EF = f̃ and SUP(EF) = SUP(f).

Proof. Let E denote the function f_{j} + τ̃_{j}. Since EF − f_{j} is u.s.c. and majorizes τ_{j}, it also majorizes τ̃_{j} = E − f_{j}, hence EF ≥ E. On the other hand, for every k the difference f_{j} − f_{k} is u.s.c. (consider the cases k > j and k ≤ j), so

E − f_{k} = f_{j} + τ̃_{j} − f_{k} is u.s.c. and
E − f_{k} ≥ f_{j} + τ_{j} − f_{k} = f − f_{k} ≥ 0.

Therefore E is a superenvelope, and E ≥ EF. The remaining statements of the assertion are now obvious. □

4. Affine functions and attainability

In this section we continue to assume the conditions 2.16 governing F and in addition we suppose the common domain K of the functions f_{k} is a Choquet simplex and the functions f_{k} are affine. We say a cover G of F is offset if g_{k} > f_{k} − f_{k−1} for every k.

Proposition 4.1. Given F and K as above, the following hold.

(4.2)  EF = inf_{G_{A}} Σ̃G_{A} = inf_{G_{A}} Σ̂G_{A} = inf_{G′_{A}} Σ̂G′_{A} = lim_{G′_{A}} Σ̂G′_{A}

where G_{A} ranges over all affine covers of F, and G′_{A} ranges over all affine offset covers of F. The u.s.c. function EF is concave.

Proof. Obviously, in the formula EF = inf_{G} Σ̃G of Theorem 2.18, we can take the infimum only over offset covers. Then, using Theorem 2.11, every such cover dominates another cover G_{A} consisting of affine functions. This proves the first equality. The second equality holds, as explained in Remarks 2.14, because the functions ΣG_{A} are affine. (This shows EF is concave.) The next equality is obvious and the last follows because, by Theorem 2.11 again, the affine offset covers form a directed family. □

Another function of interest to us in this case is

E_{A}F = inf E_{A},

defined as the infimum of all affine superenvelopes of F. This is also a concave nonnegative u.s.c. function. In fact we show the following equalities:

Theorem 4.3.

E_{A}F = EF, inf_{E_{A}} SUP(E_{A}) = SUP(EF).

Remark 4.4. Note that since the affine superenvelopes usually DO NOT form a directed family (see Example 4.7), the second equality is not a consequence of the first and a statement on exchanging suprema and infima as in Proposition 2.4.

Proof of Theorem 4.3. The inequalities “≥” are obvious and equalities hold if EF ≡ ∞. Suppose EF is bounded and let g be an affine continuous function above the sum of some fixed affine cover, g ≥ ΣG_{A}, G_{A} = (g_{k})_{k=1}^{∞}. We will find an affine superenvelope E_{A} below g. Set

E_{A} = g − (ΣG_{A} − f)

This E_{A} is an affine function not larger than g. Moreover, for each k ∈ ℕ,

E_{A} − f_{k} = (g − ΣG_{A}) + ∑_{n=k+1}^{∞}(f_{n} − f_{n−1}) = g − ∑_{n=1}^{k} g_{n} − ∑_{n=k+1}^{∞} (g_{n} − (f_{n} − f_{n−1})) ,

which is nonnegative (see middle expression) and u.s.c. (each g_{n} − (f_{n} − f_{n−1}) is l.s.c., and so is the sum as a nondecreasing limit). So, E_{A} is an affine superenvelope of F. Thus E_{A}F ≤ Σ̂G_{A}. But this is true for any ΣG_{A}, so, by (4.2), E_{A}F ≤ EF. For the second statement let g ≡ SUP(EF) + ϵ. Using Property 2.2(4) and the fact that affine offset covers are directed, we can find an affine offset cover G′_{A} with Σ̃G′_{A} < g. The above argument now produces an affine superenvelope E_{A} below SUP(EF) + ϵ, which ends the proof. □

Given x in a metrizable Choquet simplex K, let μ_{x} be the unique Borel probability on exK such that for every affine continuous (or u.s.c.) function a on K,

a(x) = ∫_{exK}(a|_{exK})dμ_{x}

where a|_{exK} is the restriction of a to exK. The point x is called the barycenter of the measure μ_{x}. The restriction map establishes a 1-1 order preserving correspondence between affine continuous functions on K and certain continuous functions on exK (namely those which are restrictions of affine continuous functions on K). We can define

g^{aff}(x) = ∫_{exK} gdμ_{x}

for any bounded continuous (or u.s.c.) function g on exK, but in general this function need not be continuous (u.s.c.), because the map x ↦ μ_{x} need not be continuous.

A Bauer simplex is a Choquet simplex K such that the extreme set exK is compact. In this case the map x ↦ μ_{x} is a continuous affine bijection onto the Borel probabilities on exK [P1, pp.65-6]. Consequently, for a Bauer simplex K the map f ↦ f|_{exK} gives a 1-1 correspondence between affine continuous functions on K and all continuous functions on exK, and the same formula establishes a 1-1 correspondence between all nonnegative affine u.s.c. functions on K and all nonnegative u.s.c. functions on exK. Consequently, for a Choquet simplex K, if g is nonnegative continuous (u.s.c.) on exK and vanishes outside a compact subset C of exK, then g^{aff} is necessarily nonnegative continuous (u.s.c.) on K (because the closed convex hull of C is a Bauer simplex with extreme set C, and elsewhere on K the function g^{aff} is 0).

The discussion of Bauer simplicies justifies the easy

Remark 4.5. Our examples 2.19, 2.20, 2.21 can be adapted to produce corresponding affine examples on Bauer simplices, by viewing the given compact domain as the extreme set of a simplex and extending all functions affinely.

If now F is a sequence of affine nonnegative u.s.c. functions on a Bauer simplex K then the restriction map is a 1-1 correspondence between all affine superenvelopes of F and all superenvelopes of F|_{exK}. But this implies that the affine superenvelopes of F form a directed family and that the smallest affine superenvelope of F exists. Combining this with Theorem 4.3 we conclude:

Theorem 4.6. If F is a sequence of nonnegative affine u.s.c. functions defined on a Bauer simplex K, then EF is affine.

In the following example we show that for general Choquet simplexes the situation may be significantly different: the smallest affine superenvelope need not exist, and the infimum of pointwise suprema of such envelopes may never be attained. (This example also shows that affine superenvelopes need not form a directed family, because the limit would equal the smallest superenvelope and it would be affine.)

Example 4.7. We thank Bob Phelps [P2] for showing us the road that leads to this example. It is an example of a sequence F of nonnegative affine u.s.c. functions defined on a Choquet simplex K for which SUP(E_{A}) > SUP(EF) (= inf_{E_{A}} SUP(E_{A}), by the second statement of Theorem 4.3), for every affine superenvelope E_{A} of F. Moreover, in the example EF = f̃, and in particular SUP(EF) = SUP(f).

Let K be the Choquet simplex whose set of extreme points consists of a point b_{1}, a sequence (a_{n})_{n≥1} converging to b_{1}, and a sequence (b_{n})_{n≥2} converging to

b = ∑_{n=1}^{∞} 2^{−n}a_{n}.

Define F = (f_{k})_{k∈ℕ}, where f_{k} = (∑_{n=1}^{k} 1_{{b_{n}}})^{aff}. Since characteristic functions of single points have compact supports, f_{1} is an affine u.s.c. function, and f_{k} − f_{k−1} is an affine continuous function if k ≥ 2. Consequently by Proposition 3.13,

EF = (1_{{b_{1}}})^{aff} + (∑_{n=2}^{∞}(1_{{b_{n}}})^{aff})^{∼} .

By Theorem 4.9(1) below, EF must assume its maximum on the closure of the extreme points. The maximum of EF on exK is 1. The only new accumulation point of exK is b, and since (1_{{b_{1}}})^{aff}(b) = 0 we have

EF(b) = 0 + (∑_{n=2}^{∞}(1_{{b_{n}}})^{aff})^{∼}(b) ≤ 0 + 1 .

Consequently SUP(EF) = 1.

Now suppose E_{A} is an affine superenvelope of F not exceeding 1. Clearly, E_{A}(b_{n}) = 1 for each n. By upper semi-continuity, E_{A}(b) = 1. Now, E_{A} restricted to the Bauer simplex whose extreme set I consists of the sequence (a_{n}) and its limit b_{1}, is an affine u.s.c. function, so it is E^{aff}, where E is a u.s.c. function on I. Since E_{A} ≤ 1, also E ≤ 1, and because

1 = E_{A}(b) = ∑_{n=1}^{∞} 2^{−n}E(a_{n}),

E equals 1 at all points a_{n} (n ≥ 1). By upper semi-continuity, E(b_{1}) = 1. But now E_{A} − f_{1} is not u.s.c., because it has a “bad” jump at b_{1}. This concludes the example.

Example 4.8. Here is an example of F such that SUP(EF) > sup{EF(μ): μ ∈ exK}. Consider a simplex whose extreme points consist of two elements b_{1}, b_{2} and a sequence (a_{n})_{n≥2} converging to b = \frac{1}{2}(b_{1} + b_{2}). Set f_{1} = 1_{\{b_{1},b_{2}\}}^{aff} (so that f_{1}(b) = 1), and f_{k} = f_{k−1} + 1_{{a_{k}}}^{aff} (k ≥ 2). Using Proposition 3.13, it is easy to verify SUP(EF) = 2 (attained at b as in Example 2.19) and EF = 1 at all extreme points.

Finally, we show the maximum of EF must be achieved on the closure of the extreme points, and we extend the estimate of Proposition 3.10 to the case of an affine sequence F defined on a Choquet simplex.

Theorem 4.9. Suppose f is bounded. Then (1) SUP(EF) = sup{EF(μ): μ ∈ \overline{exK}\}, and (2) if x ∈ exK and r(x) denotes its order within the set \overline{exK}, then EF(x) ≤ f(x) + r(x)u(x).

Remark 4.10. Unfortunately, the last inequality does not apply in general to x ∈ \overline{exK}, and therefore does not provide an upper bound for SUP(EF).

Proof. By Proposition 3.10, both assertions hold on Bauer simplices, where (EF)|_{exK} equals E(F|_{exK}). For a general Choquet simplex K let M denote the Bauer simplex of all probability measures supported by \overline{exK}. The map π assigning to each such measure its barycenter is an affine continuous map from M onto K. Let F′ denote the lift of F to M. If E is a superenvelope of F′, then the pushed down function E^{[K]} is a superenvelope of F, and conversely a superenvelope of F lifts to a superenvelope of F′. Therefore EF = (EF′)^{[K]} and

SUP(EF) ≥ SUP(EF|\overline{exK}) = SUP((EF′)^{[K]}|\overline{exK})
≥ SUP(EF′|_{exM}) = SUP(EF′) = SUP(EF) .

This proves (1). Now suppose x ∈ exK, so x has a unique preimage x′ in M, and x′ ∈ exM. Then

EF(x) = (EF′)^{[K]}(x) = (EF′)(x′) ≤ h(x) + r(x)u(x)

where the passage from x′ to x in the last equality holds because the restriction of π to exM is a homeomorphism onto \overline{exK}. □

5. The Sex Entropy Theorem

We now turn to symbolic extensions. For a self contained statement of our key result, we recall some notation.

Definitions 5.1. We let K(X, T) denote the space of T -invariant Borel probabilities on X. We let φ: (Y, S) → (X, T) denote a symbolic extension (i.e. φ is a continuous surjection Y → X intertwining S and T, and (Y, S) is a subshift over a finite alphabet). The extension entropy function of φ is by definition the function h_{ext}^{φ}: K(X, T) → [0, +∞) which sends μ to max{h(S, ν): ν ∈ K(Y, S) and φν = μ}. Given (X, T), we will say a sequence of partitions P_{k} of X is refining if the maximum diameter of an element of P_{k} goes to zero with k and for each k the partition P_{k+1} refines P_{k}. The partitions have small boundaries if their boundaries have measure zero for all μ in K(X, T).

Definition 5.2. An entropy structure for (X, T) is a sequence H of functions h_{k} defined on K(X, T) in one of two ways.

(I) Suppose P_{k} is a refining sequence of finite Borel partitions with small boundaries. Define H by setting h_{k}: μ ↦ h(T, μ, P_{k}).

(II) Suppose (Z, R) is an aperiodic minimal zero entropy system, λ ∈ K(Z, R), and (X × Z, T × R) is the product system. Define H by first choosing an entropy structure H′ = (h′_{k}) for (X × Z, T × R) defined as in (I), and then setting h_{k}(μ) = h′_{k}(μ × λ). (The existence of such a structure H′ follows from Fact 7.6(3).)

Remark 5.3. Not every system admits a refining sequence of finite Borel partitions with small boundaries, so it is tempting to take only (II) as a definition for entropy structure, with some fixed choice for (Z, R) such as the dyadic adding machine. However, this would not produce a canonical sequence H (H still depends on the choice of partition), and (I) is more direct in the many cases where it applies. The supply of sequences providing an entropy structure satisfying Theorem 5.5 will be greatly enriched in [D3].

Remark 5.4. In place of 5.2(II), it is natural to consider defining an entropy structure on K(X, T) simply by pushing down to K(X, T) (by supremum over preimages) the entropy structure H′ of (X × Z, T × R). This would produce affine u.s.c. functions on K(X, T) which converge nondecreasingly to the entropy function. We did not make this choice because we also want the functions to have u.s.c. differences, and this need not be preserved by pushing down.

We now state our key result.

Theorem 5.5 (Sex Entropy Theorem). Suppose T is a homeomorphism of a compact metrizable space X with entropy structure H = (h_{k}), and h_{top}(T) < ∞. Suppose E: K(X, T) → [0, +∞). Then the following are equivalent.

(1) E is a bounded affine superenvelope of H.

(2) There is a symbolic extension φ: (Y, S) → (X, T) such that E = h_{ext}^{φ}.

The proof of Theorem 5.5 splits into two parts. In the next section, we will prove the theorem for a convenient choice of H when the space X has dimension zero. Then in Section 7, we will address general (X, T) and alternate choices of H.

6. Proof of the Sex Entropy Theorem: a zero dimensional case

This section is devoted to proving Theorem 5.5 in the following special case:

Case Z 6.1. X is zero dimensional and the entropy structure is given by 5.2(I) with each P_{k} a partition into nonempty clopen sets.

Notation 6.2. We now set some notation for the rest of this section. From the given clopen partitions P_{k}, we fix a presentation of (X, T) as the associated inverse limit of subshifts,

(X, T) = \overleftarrow{lim_{k→∞}}(X_{k}, S) .

The alphabet for X_{k} is some set in bijective correspondence with P_{k}. The points of X_{k} correspond to itineraries through P_{k} under T. The bonding maps (factor maps) X_{k+1} → X_{k} are the amalgamations (surjective one-block maps) which reflect inclusion of an element of P_{k+1} in a unique element of P_{k}. The entropy structure H has an equivalent description here as h_{k}(μ) = h(S, μ_{k}), where μ_{k} denotes the projection of μ to the system (X_{k}, S). Of course, lim_{k} h_{k}(μ) = h(μ), the entropy of T with respect to the measure μ. The functions h_{k} are nonnegative, u.s.c., affine and bounded by h(T). As explained in [D2], and below, they also have u.s.c. differences.

Measure-theoretic conditional entropy.

Recall that a sequence (a_{n}) of real numbers is called subadditive if a_{n+m} ≤ a_{n} + a_{m} for all n, m ∈ ℕ. For such a sequence (a_{n}), the sequence (a_{n}/n) converges to its infimum. Moreover, if (n_{i}) is a subsequence satisfying n_{i}|n_{i+1} then the convergence is monotone along (n_{i}). By the way, we recall that the property n_{i}|n_{i+1} is equivalent to (n_{i}) being a base of an odometer.

We recall some elementary entropy theory. For details and proofs see e.g. [Wa]. Let (Y, ν, S) be a measure preserving system, and let P_{1} and P_{2} be two partitions

of Y, P_{1} ≽ P_{2} (P_{1} a refinement of P_{2}). We denote

P_{1}^{n} := ⋁_{i=0}^{n−1} T^{−i}P_{1}.

Throughout this paper we shall write:

\begin{matrix}H(ν, P_{1}) = − ∑_{A∈P_{1}} ν(A)log(ν(A)); \\ H(ν, P_{1}|P_{2}) = ∑_{B∈P_{2}} ν(B)H(ν_{B}, P_{1}) = H(ν, P_{1}) − H(ν, P_{2}), \end{matrix}

where ν_{B} denotes the conditional measure induced by ν on B (or zero if ν(B) = 0). It is known that H(ν, P_{1}^{n}|P_{2}^{n}) is a subadditive sequence, and hence divided by n converges to its infimum h(ν, P_{1}|P_{2}) = h(ν, P_{1}) − h(ν, P_{2}) known as the conditional entropy. We denote

H_{n}(ν, P_{1}|P_{2}) = \frac{1}{n}H(ν, P_{1}^{n}|P_{2}^{n}),

and we call it the (nth) approximative conditional entropy.

We can now recover a central observation of [D2]:

Proposition 6.3. If φ is a factor map (quotient map) of subshifts, φ: (Y, S) → (Y′, S), then the conditional entropy map ν ↦ h(ν) − h(φν) is u.s.c.

Proof. Let P_{2} be the pullback of the zero coordinate partition of Y′, and let P_{1} be a generating clopen partition of Y which refines P_{2}. Then the function H_{n}: ν ↦ H_{n}(ν, P_{1}|P_{2}) is continuous. Therefore inf_{n} H_{n} is u.s.c. This infimum is the conditional entropy map. □

It follows that for our entropy structure H, the difference functions h_{k} − h_{k−1} are u.s.c. We can now prove the easy direction of the Sex Entropy Theorem.

Proof of Theorem 5.5 (2) ⟹ (1) in Case Z.

For the given symbolic extension φ: (Y, S) → (X, T), we must show h_{ext}^{φ} is a bounded affine superenvelope of H. Clearly h_{ext}^{φ} is bounded and each h_{ext}^{φ} − h_{k} is nonnegative. For each k, postcomposing φ with the projection X → X_{k} produces a factor map φ_{k} of subshifts. Because

(h_{ext}^{φ} − h_{k})(μ) = max{h(ν) − h(φ_{k}ν) : φν = μ, ν ∈ K(Y, S)}

we have that h_{ext}^{φ} − h_{k} is the pushdown (by φ) of a function which is u.s.c. (by Proposition 6.3) and affine. It follows that h_{ext}^{φ} − h_{k} is u.s.c., and it is affine by Proposition 2.7 because the map φ sends ergodic measures to ergodic measures. □

Now we set some notation for the next lemma, which will be essential to our proof of the converse. Let (Y, S) and (Y′, S) be full shifts: Y = Λ^{ℤ} and Y′ = Λ′^{ℤ}, where Λ and Λ′ are finite sets (called alphabets). Suppose also that (Y′, S) is a factor of (Y, S) via an amalgamation π. This means that the “zero coordinate partition” P_{1} of Y into cylinders of the form U_{a} = {y ∈ Y : y(0) = a} (a ∈ Λ) is a refinement of the corresponding “zero coordinate partition” P_{2} of Y′ lifted to Y by preimage. In this case H_{n}(ν, P_{1}|P_{2}) is determined by observing the (cylinders generated by) blocks of length n in Y′ and their preimages (which are blocks of length n in Y). Because P_{1} and P_{2} generate the Borel sigma-fields in Y and Y′, respectively, they suffice for all evaluations of entropy, and we will not use any other partitions. We therefore skip their indicators in the notation of h(ν, P_{1}), h(ν, P_{2}), h(ν, P_{1}|P_{2}) and H(ν, P_{1}|P_{2}). Instead, we will write h(ν), h(πν), h(ν|πν) and H_{n}(ν|πν), respectively, where πν always denotes the image of ν on Y′ defined by πν(A) = ν(π^{−1}(A)). Now, suppose C is a block of some length m in Y. Its image by π is a block B in Y′ of the same length. Consider the invariant measure ν_{C} carried by the periodic orbit of the sequence … CCC . . .. The code π sends this measure to the measure μ_{B} supported by … BBB . . .. In such a case, we define

H_{n}(C|B) = H_{n}(ν_{C}|μ_{B}).

Lemma 6.4. Let (Y, S) and (Y′, S) be full shifts, Y′ = Λ′^{ℤ} and Y = Λ^{ℤ}. Let π: Y → Y′ be an amalgamation. Denote λ = #Λ. For n, m ∈ ℕ and blocks C ∈ Λ^{m} and B ∈ Λ′^{m}, B = π(C), let H_{n}(C|B) denote the corresponding approximative conditional entropy. Then for every n ∈ ℕ and ϵ > 0 there exists an m_{(n,ϵ)} ∈ ℕ such that for every m ≥ m_{(n,ϵ)} the following holds

∑_{C∈π^{−1}(B)} exp[−mH_{n}(C|B)] ≤ me^{mϵ}(log λ).

Proof. Consider π^{−1}(B) as a probability space with uniform probability P (C) = \frac{1}{#π^{−1}(B)} = p ≥ λ^{−m}, with a random variable X(C) = H_{n}(C|B). The expression to estimate equals \frac{1}{p}E(e^{−mX}) (here E denotes the expected value). By Lemma 2 in [D2] (which is simply a conditional version of Lemma 1 in [BGH]), we have for every t ≥ 0 that

#{C ∈ π^{−1}(B): H_{n}(C|B) ≤ t} ≤ e^{m(t+ϵ)}

for m sufficiently large. Clearly, the above cardinality does not exceed #π^{−1}(B), and it is zero for t < 0. Eventually, we can write

p#\{C ∈ π^{−1}(B): H_{n}(C|B) ≤ t\} ≤ \begin{cases}0 , & t < 0; \\ min(pe^{m(t+ϵ)}, 1) , & t ≥ 0. \end{cases}

This can be interpreted as saying that the cumulative distribution function of X, F_{X}(t) = Prob(X ≤ t), is below the cumulative distribution function of W, where W is a random variable with density

f(t) = pme^{m(t+ϵ)}

on the interval [0, \frac{− log p}{m} − ϵ], and with an atom of mass pe^{mϵ} at zero. This means that the mass of probability of the distribution of X is moved to the right compared to that of W. Because the function e^{−mx} is decreasing, the expected value E(e^{−mX}) is not larger than

E(e^{−mW}) = pme^{mϵ}(\frac{− log p}{m} − ϵ) + 1 · pe^{mϵ} ≤ pme^{mϵ}(log λ + \frac{1}{m} − ϵ).

This provides the required estimate of \frac{1}{p}E(e^{−mX}). □

Topological fiber entropy.

For the final entropy calculation of our theorem (Stage 4), we will need a variational principle which lets us make the transition from word counts to the sex entropy of a measure. To recall this principle, let (X, T) and (Y, S) be two topological dynamical systems and let π: Y → X be a topological factor map. For an open cover A of Y and an x in X we set

H(A|x) = log min{#F: F ⊂ A, ⋃ F ⊃ π^{−1}(x)}

and for a probability measure μ on X we set

H(A|μ) = ∫ H(A|x)dμ(x) .

The function x ↦ H(A|x) is easily seen to be u.s.c. on X and hence also μ ↦ H(A|μ) is a u.s.c. function on the space of measures, and it is obviously affine. Moreover, if the cover A is a partition into closed and open sets, then these maps are continuous. If μ is an invariant measure then, by a subadditivity argument, the sequence \frac{1}{n}H(A^{n}|μ) (where A^{n} denotes the joint cover ⋁_{i=0}^{n−1} S^{−i}(A)) converges to its infimum and the convergence is nonincreasing along any subsequence (p_{k})_{k∈ℕ} such that for each k, p_{k+1} is a multiple of p_{k}. Thus, on K(X, T) we have a well defined affine u.s.c. function

h(A|·) = lim_{n} \frac{1}{n}H(A^{n}|·)

which we call the topological fiber entropy function on K with respect to A. Finally, on K we define the function

h(Y |·) = sup_{A} h(A|·)

which we call the topological fiber entropy function. We will need the following fact: for μ ∈ K(X, T),

(6.5)  h(Y |μ) = sup_{ν}{h(ν|μ): ν ∈ K(Y, S) and πν = μ} .

In other words, h(Y |μ) = h_{ext}^{π}(μ) − h(μ). This fact is essentially a special case of the “Relativised Variational Principle” [LeWa, Theorem 2.1]. It is also the “Inner Variational Principle”, which holds for not necessarily metrizable systems [DS1, Theorem 4]. In the special case that (Y, S) is a subshift, it is known that the supremum in the definition of h(Y |·) is attained for the zero coordinate partition A (which is also a cover). We summarize what we will need for our entropy computation in the following lemma.

Lemma 6.6. Suppose φ: (Y, S) → (X, T) is a factor map, (Y, S) is a subshift, and A is the zero coordinate partition of Y. Then for μ ∈ K(X, T),

h_{ext}^{φ}(μ) = lim_{n} \frac{1}{n}H(A^{n}|μ) + h(μ)

and the functions μ ↦ H(A^{n}|μ) are continuous on K(X, T).

We are now ready for the difficult direction of the Sex Entropy Theorem.

Proof of Theorem 5.5 (1) ⟹ (2) in Case Z.

Given the bounded affine superenvelope E, we will construct a symbolic extension φ such that h_{ext}^{φ} = E. The construction is a combination and simplification of methods from [BFF] and [D2] which at the same time allows for better control over entropy properties of the extension. This new construction splits into two main stages. From an affine superenvelope we derive an SWO, a simplified version of the notion of a word oracle introduced in [BFF]; then, from the SWO we construct a symbolic extension with the desired sex entropy. In fact, we present these steps in the reversed order as stages 2 and 3.

STAGE 1. Preliminary reshaping

As in [D2], we begin with the observation, that it is convenient to consider the direct product of (X, T) with an odometer. Let p = (p_{k})_{k∈ℕ} be a sequence of positive integers such that p_{k+1} = p_{k}q_{k} with q_{k} being an integer larger than 1, for each k. Let (G_{p}, τ) denote the rotation by 1 of the odometer to base p. We denote (X′, T′) = (X × G_{p}, T × τ).

We represent (X, T) = \overleftarrow{lim}(X_{k}, S) as the shift transformation on a closed shiftinvariant family of infinite matrices

x = (x_{k,n})_{k∈ℕ,n∈ℤ},

such that each row (indexed by k) is a sequence over some finite alphabet Δ_{k}. We let

x_{k} = (x_{s,n})_{s≤k,n∈ℤ}

denote the projection of x onto the first k rows. In this setting, (X_{k}, S) is the symbolic system obtained from (X, T) by the projection onto the first k rows. The alphabet of X_{k} is contained in the product Λ_{k} = Δ_{1} × ⋯ × Δ_{k}. Note that each projection from X_{k+1} to X_{k} is an amalgamation.

The product (X′, T′) is realized in such manner that in each row x_{k} we apply a p_{k}-periodically repeating marker, say a comma, and the positions of the commas in the (k+1)st row coincide with the positions of some (every q_{k}th) commas in the kth row. (Formally, the alphabet Δ′_{k} of the kth row is Δ_{k} ∪ Δ_{k} × {,}). Now, the projection from X′_{k} to X_{k} (erasing the commas in the first k rows) induces a map from K′_{k} = K(X′_{k}, S) onto K_{k} = K(X_{k}, S) and even though this map is not 1-1, it is obvious that h(μ′_{k}) = h(μ_{k}) whenever μ′_{k} projects to μ_{k}. Thus the conditional entropy functions on K′ = K(X′, T′) coincide with the functions h_{k} lifted from K = K(X, T). So, if E_{A} is a bounded affine superenvelope for H on K, its lift (also denoted by E_{A}) becomes a bounded affine superenvelope for the entropy sequence (also denoted by H) on K′. So, it suffices to construct a symbolic extension of (X′, T′) with the required sex entropy function. From now on we will use (X, T) to denote (X′, T′), we will also skip the apostrophes in X′_{k}, K′_{k}, etc. We are free to choose the base p according to our needs, and we will specify p in Stage 3.

For given k, the blocks B of length p_{k} over the alphabet Δ_{k} ending with a comma, will be called k-blocks and their collection will be denoted by B_{k}. A rectangular p_{k} × k block R having a concatenation of \frac{p_{k}}{p_{s}} of s-blocks in the sth row (for each s = 1, 2, … k) will be called an k-rectangle, and the collection of such will be denoted by R_{k}. Notice that a (k+1)-rectangle is a concatenation of q_{k} k-rectangles with a (k+1)-block added as a new row, which we denote by

R = (\begin{matrix}R^{(1)}R^{(2)} … R^{(q_{k})} \\ B \end{matrix})

(see figure below; stars replace any symbols from any Δ_{k}).

\begin{matrix}∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, & ∗ ∗ ∗, \\ ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗, & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗, & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗, \\ ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗ & ∗ ∗ ∗, \end{matrix}

Also note that a k-rectangle is in fact a block over Λ_{k} (with added commas). Let R_{k}(X) denote the collection of all k-rectangles that actually occur in X.

STAGE 2. Simplified word oracle.

By a simplified word oracle (SWO) we shall understand a sequence of functions α_{k}: R_{k} ↦ ℕ such that α_{k}(R) = 0 if and only if R ∉ R_{k}(X), and

(6.7)  α_{k}(R^{(1)})α_{k}(R^{(2)}) ⋯ α_{k}(R^{(q_{k})}) ≥ ∑_{B∈B_{k+1}} α_{k+1} (\begin{matrix}R^{(1)}R^{(2)} … R^{(q_{k})} \\ B \end{matrix}) .

We now describe a construction of a symbolic extension (Y, S) of (X, T) given an SWO. Let \overline{X}_{k} denote the set of all possible infinite concatenations of k-rectangles from R_{k}(X). Clearly, X_{k} ⊂ \overline{X}_{k}. Our starting point will be constructing (in a consistent way), for each k, a certain symbolic extension (\overline{Y}_{k}, S) of (\overline{X}_{k}, S) such that precisely α_{k}(R) blocks will be seen above each k-rectangle R.

Note that R_{1} = B_{1}. Let λ ∈ ℕ be such that λ^{p_{1}} ≥ ∑_{R∈R_{1}} α_{1}(R), and let Λ_{Y} be an alphabet with cardinality λ. Step 1. For each R ∈ R_{1} pick a family F_{1}(R) of α_{1}(R) different blocks of length p_{1} over Λ_{Y} (for R not appearing in X this family is empty). The cardinality of Λ_{Y} allows to do it so that for different blocks R the families F_{1}(R) are disjoint. Let F_{1} be the union of all families F_{1}(R). Attach a comma at the end of every block in F_{1}. By disjointness, we can define a map ρ_{1} from F_{1} onto R_{1}(X) sending each C to the unique R for which C ∈ F_{1}(R). Let \overline{Y}_{1} be the set of all possible concatenations of blocks from F_{1}. Clearly, ρ_{1} sends \overline{Y}_{1} onto \overline{X}_{1} and what can be seen above each 1-rectangle R is F_{1}(R) whose cardinality is exactly α_{1}(R). Step k+1. Suppose the task has been completed for some k. Consider a concatenation R^{(1)}R^{(2)} … R^{(q_{k})} of some k-rectangles. Above it in \overline{Y}_{k}, exactly as many as α_{k}(R^{(1)})α_{k}(R^{(2)}) ⋯ α_{k}(R^{(q_{k})}) blocks can be seen: namely all possible concatenations of q_{k} blocks: first from F_{k}(R^{(1)}), next from F_{k}(R^{(2)}), and so on. By the inequality (6.7) in the definition of an SWO, it is thus possible, for each (k +1)-

rectangle R = (\begin{matrix}R^{(1)}R^{(2)} … R^{(q_{k})} \\ B \end{matrix}), to disjointly select from these concatenations

a family F_{k+1}(R) of cardinality α_{k+1}(R). Note that if two (k+1)-rectangles differ already in the the first k rows then their families are chosen from disjoint collections of concatenations, so obviously they are disjoint. Let F_{k+1} be the union of all so selected families (over all (k +1)-rectangles R, after removing all but the terminal comma in every such block). Let ρ_{k+1} be the map from F_{k+1} onto R_{k+1}(X) sending each C to the unique R for which C ∈ F_{k+1}(R). Let \overline{Y}_{k} be the set of all possible concatenations of blocks from F_{k+1}. Clearly, ρ_{k+1} sends \overline{Y}_{k+1} onto \overline{X}_{k+1} and what can be seen above each (k+1)-rectangle R is F_{k+1}(R) whose cardinality is exactly α_{k+1}(R). The task has been now completed for k+1.

Notice that there are natural maps π_{k}: \overline{X}_{k+1} → \overline{X}_{k} (by projection onto the first k rows, hence they coincide on X_{k+1} with the bonding maps of the inverse limit defining X). These maps are not surjective, but it will not be essential. Also, we have natural (not surjective) maps κ_{k}: \overline{Y}_{k+1} → \overline{Y}_{k}. Namely, each y ∈ \overline{Y}_{k+1} being a concatenation of blocks from F_{k+1} is, at the same time, a concatenation of smaller blocks from F_{k}, hence, adding appropriate commas, we can view y as an element of \overline{Y}_{k}. Now notice that the diagram below commutes:

\overline{Y}_{1} ⟵_{κ_{1}} \overline{Y}_{2} ⟵_{κ_{2}} \overline{Y}_{3} ⟵_{κ_{3}} …
↓^{ρ_{1}}  ↓^{ρ_{2}}  ↓^{ρ_{3}}
\overline{X}_{1} ⟵_{π_{1}} \overline{X}_{2} ⟵_{π_{2}} \overline{X}_{3} ⟵_{π_{3}} …

The above diagram defines a factor map ρ from the inverse limit (\overline{Y} , S) = \overleftarrow{lim}(\overline{Y}_{k}, S) onto the inverse limit (\overline{X}, S) = \overleftarrow{lim}(\overline{X}_{k}, S). Clearly X ⊂ \overline{X} and we define Y to be the preimage by ρ of X. Because all maps κ_{k} do not alter the symbols of y_{k+1} (only more commas are added), (\overline{Y} , S) (and hence also (Y, S)) can be viewed as a subsystem of the direct product of a symbolic system (over the alphabet Λ_{Y}) with the odometer to base p. Finally, the odometer can be replaced by a zero entropy strictly ergodic symbolic extension, for example, by a regular Toeplitz flow of which the odometer is a factor [JK]. This completes the construction of a symbolic extension of (X, T) using a given SWO.

We have the following convergence:

Lemma 6.8. For μ ∈ K and μ_{k} denoting the projection of μ to K_{k},

(6.9)  h(Y |μ) = lim_{k} h(\overline{Y}_{k}|μ_{k}).

Proof. Given k ∈ ℕ, we will justify the following claims:

h(Y |μ) = sup{h(ν) − h(μ) : ν ∈ ρ^{−1}(μ)}
≤ sup{h(κ_{k}(ν)) − h(μ_{k}) : ν ∈ ρ^{−1}(μ)}
≤ sup\{h(ν_{k}) − h(μ_{k}) : ν_{k} ∈ ρ_{k}^{−1}(μ_{k})\} = h(\overline{Y}_{k}|μ_{k})
= max{h(ν_{k}) − h(μ_{k}) : ν_{k} ∈ ρ_{k}^{−1}(μ_{k})} .

The first equality follows from the variational fact (6.5), because ρ(ν) = μ implies ν is supported on Y. The first inequality holds because for all k we have h(μ_{k}) ≤ h(μ) and also h(κ_{k}(ν)) = h(ν) (the maps κ_{k} are “essentially” embeddings, only adding periodic commas). For the second inequality, we take a supremum over a larger set. The second equality is again (6.5). The supremum is achieved by Proposition 6.3.

Similarly, h(\overline{Y}_{k}|μ_{k}) is nonincreasing in k. We conclude h(Y |μ) ≤ lim_{k} h(\overline{Y}_{k}|μ_{k}). For the converse inequality, let M_{k} denote the set of measures ν_{k} in ρ_{k}^{−1}(μ_{k}) such

that h(ν_{k}) − h(μ_{k}) ≥ lim_{ℓ} h(\overline{Y}_{ℓ}|μ_{ℓ}). These sets are nonempty by the mentioned monotonicity, since some measure ν_{k} satisfies h(ν_{k})−h(μ_{k}) = h(\overline{Y}_{k}|μ_{k}). These sets are also closed, by upper semi-continuity of h in symbolic systems, and κ_{k}(M_{k+1}) ⊂ M_{k}. By compactness, we can thus find a measure ν on ∏_{k} \overline{Y}_{k} such that κ_{k}(ν) ∈ M_{k} for each k. This measure ν is supported on Y, so h(Y |μ) ≥ h(ν) − h(μ) ≥ lim_{k} h(\overline{Y}_{k}|μ_{k}). □

STAGE 3. Constructing an SWO from an affine superenvelope.

Let E_{A} be an affine superenvelope of H. By definition the functions E_{k} := E_{A} − h_{k} are nonnegative affine and u.s.c. They converge nonincreasingly to the function E_{A} − h, which therefore is also nonnegative, affine and u.s.c.

Before we continue with the proof, we need one more technical lemma:

Lemma 6.10. Define functions E_{k}^{[K_{k}]} on K(X, T) as follows:

E_{k}^{[K_{k}]}(μ) = sup_{ξ∈K: ξ_{k}=μ_{k}} E_{k}(ξ) .

Then these are affine u.s.c. functions and lim_{k} E_{k}^{[K_{k}]} = lim_{k} E_{k} = E_{A} − h.

Proof. Given k, the function E_{k}^{[K_{k}]} is defined by supremum over preimages under the natural projection K(X, T) → K(X_{k}, S). This projection is a continuous affine surjection between Choquet simplices, and it sends extreme points to extreme points because it sends ergodic measures to ergodic measures. Because E_{k} is affine, the affinity of E_{k}^{[K_{k}]} then follows from Proposition 2.7.

Clearly the sequence E_{k}^{[K_{k}]} converges pointwise, as a nonincreasing sequence of nonnegative functions, and lim_{k} E_{k}^{[K_{k}]} ≥ lim_{k} E_{k}. So, only one inequality for the limits requires verification. Suppose

lim_{k} E_{k}^{[K_{k}]}(μ) > lim_{k} E_{k}(μ) + ϵ.

For each k we can pick a measure ξ^{k} in K projecting to μ_{k} such that E_{k}^{[K_{k}]}(μ) = E_{k}(ξ^{k}). It is seen that the ξ^{k} approach μ as k → ∞, because they agree with μ on finer and finer clopen partitions. Further, for s < k, we have E_{s}(ξ^{k}) ≥ E_{k}(ξ^{k}). Leaving s fixed and letting k → ∞, we obtain, by upper semi-continuity, E_{s}(μ) ≥ lim_{k} E_{k}(ξ^{k}) > lim_{k} E_{k}(μ) + ϵ, holding for each s, which is impossible. □

We can now continue with the main proof. By Definition 2.1(2) and Theorem 2.11, we can represent each function E_{k}^{[k]} as a strictly decreasing limit of affine continuous functions f_{k,i} defined on K_{k} and then lifted to K. Moreover, since E_{k}^{[k]} is nonincreasing in k, we can easily arrange the double sequence f_{k,i} to be nonincreasing also in k for each i. Then the sequence of functions f_{k} := f_{k,k} satisfies:

(a) f_{k} > E_{k}^{[K_{k}]}, for each k ∈ ℕ, and

(b) lim_{k} f_{k} = E_{A} − h. Because f_{k} − (h_{k+1} − h_{k}) > E_{k} − (h_{k+1} − h_{k}) = E_{k+1} and the function on the left is constant on the fibers of the projection K → K_{k+1}, we have

f_{k} − (h_{k+1} − h_{k}) > E_{k+1}^{[K_{k+1}]}.

Using the Sandwich Theorem 2.8 (on K_{k+1}), we can find a continuous function f′_{k+1} with f_{k} − (h_{k+1} − h_{k}) > f′_{k+1} > E_{k+1}^{[K_{k+1}]}. Then, min(f_{k+1}, f′_{k+1}) is a continuous function strictly above E_{k+1}^{[K_{k+1}]}, hence, by Theorem 2.11, we can find an affine continuous function f′′_{k+1} such that f_{k} −(h_{k+1} −h_{k}) > f′′_{k+1} > E_{k+1}^{[K_{k+1}]}. We redefine f_{k+1} to equal f′′_{k+1}. We do this for k = 1 and we proceed inductively.

In the end, the conditions (a) and (b) still hold, and f_{k} − (h_{k+1} − h_{k}) > f_{k+1}. Appealing to upper semicontinuity of h_{k+1} − h_{k}, we choose a sequence (ϵ_{k}) of positive numbers decreasing to zero and satisfying

(c) f_{k} − f_{k+1} − 5ϵ_{k} > h_{k+1} − h_{k} .

Next, we will choose the base p = (p_{k}) of an odometer, i.e., a sequence satisfying p_{k}|p_{k+1}, growing rapidly enough to satisfy several conditions. Namely, note that if (n_{i}) is such a base then, for each k, the functions H_{n_{i}}(μ_{k+1}|μ_{k}) converge nonincreasingly to h_{k+1} − h_{k}, and thus by (c) the continuous functions μ ↦ max{f_{k} − f_{k+1} − 5ϵ_{k}, H_{n_{i}}(μ_{k+1}|μ_{k})} converge nonincreasingly to the continuous function f_{k} − f_{k+1} − 5ϵ_{k}. By Property 2.2(4) this convergence is uniform, so we may choose p_{k} to be an index n_{i} large enough to guarantee

(d) (f_{k} − f_{k+1})(μ) − 4ϵ_{k} > H_{p_{k}}(μ_{k+1}|μ_{k}) for all μ and we also require

(e) p_{k+1} ≥ m_{(p_{k},ϵ_{k})} of Lemma 6.4 applied to the amalgamation from X_{k+1} to X_{k},

(f) p_{k+1}e^{−p_{k+1}ϵ_{k}} log(#Λ_{k+1}) ≤ 1, and

(g) e^{p_{k+1}(b+ϵ_{k})} ≥ ⌈e^{p_{k+1}b}⌉ whenever b ≥ 0, where ⌈·⌉ denotes the integer ceiling. (It suffices to require e^{p_{k+1}ϵ_{k}} ≥ 2.)

Observe that K_{k+1} is a subset of the set of all shift invariant measures of the full shift over the alphabet Λ_{k+1}. The functions h_{k+1} − h_{k} and μ ↦ H_{p_{k+1}}(μ_{k+1}|μ_{k}) are well defined (as entropy) and u.s.c. on this larger set. For each k we find a continuous affine extension of f_{k} to measures on the full shift over the alphabet Λ_{k}, and by lifting we also have f_{k} defined on measures on the full shift over the alphabet Λ_{k+1}. Then (d) holds on some open neighborhood U_{k+1} of K_{k+1}.

Also note that the periodic measure μ_{B} carried by the orbit of … BBB …, where B is any sufficiently long block appearing in X_{k}, is in U_{k}. As explained prior to Lemma 6.4, we will identify the block with the measure, so that h_{k}(B), f_{k}(B), etc. have a definite meaning. Because a subsequence of a base of an odometer is again a base of an odometer, and the inequalities (d) – (g) are satisfied for subsequences of (p_{k}), we may specify two more requirements on the speed of growth of the numbers p_{k}. First, take the length p_{k} of a k-rectangle so large that

(h) if R is a k-rectangle, then μ_{R} is in U_{k}. Then, by appeal to f_{k} being continuous affine, we take p_{k} large enough that the value of f_{k} on any concatenation R^{(1)}R^{(2)} … R^{(q)} of k-rectangles is close to the corresponding convex combination of values:

(i)  |f_{k}(R^{(1)}R^{(2)} … R^{(q)}) − \frac{1}{q} ∑_{i=1}^{q} f_{k}(R^{(i)})| < ϵ_{k} .

We can now define the SWO. For a k-rectangle R appearing in X, let

α_{k}(R) = ⌈e^{p_{k}f_{k}(R)}⌉ .

We need to verify the condition (6.7) in the definition of an SWO. Let C = R^{(1)}R^{(2)} … R^{(q_{k})} be a concatenation of k-rectangles appearing in X. If C does not occur in X, then the right side of (6.7) is zero and the inequality (6.7) holds trivially; so suppose C occurs in X. Then, using (g), (d), and (i) for the consecutive inequalities, we can write

∑_{B∈B_{k+1}} α_{k+1} (\begin{matrix}C \\ B \end{matrix}) = ∑_{B∈B_{k+1}} ⌈exp [p_{k+1}f_{k+1} (\begin{matrix}C \\ B \end{matrix})]⌉

≤ ∑_{B∈B_{k+1}} exp [p_{k+1} (f_{k+1} (\begin{matrix}C \\ B \end{matrix}) + ϵ_{k})]

≤ ∑_{B∈B_{k+1}} exp [p_{k+1} (f_{k}(C) − H_{p_{k}} ((\begin{matrix}C \\ B \end{matrix}) |C) − 3ϵ_{k})]

≤ exp [\frac{p_{k+1}}{q_{k}} ∑_{i=1}^{q_{k}} f_{k}(R^{(i)})] e^{−2p_{k+1}ϵ_{k}} ∑_{B∈B_{k+1}} exp [−p_{k+1}H_{p_{k}} ((\begin{matrix}C \\ B \end{matrix}) |C)] .

The first term in the last expression above is not larger than

α_{k}(R^{(1)})α_{k}(R^{(2)}) ⋯ α_{k}(R^{(q_{k})}).

By (e) and Lemma 6.4, the remaining part of that expression is not larger than p_{k+1}e^{−p_{k+1}ϵ_{k}} log(#Λ_{k+1}), which by (f) is at most 1. This verifies (6.7).

STAGE 4. Entropy calculation

It remains to verify h_{ext}^{φ} = E_{A}. First we compute h(\overline{Y}_{k}|μ_{k}) for measures μ_{k} carried by \overline{X}_{k}. To begin, consider a periodic measure μ_{C}, where C = R^{(1)}R^{(2)} … R^{(ql)} is a concatenation of ql k-rectangles (q, l ∈ ℕ). Let B(i, n, C) denote the set of blocks of length n occurring as the word y[i, i + n − 1] for some point y such that (ρ_{k}(y))[jqlp_{k}, (j + 1)qlp_{k} − 1] = C for all j (i.e., ρ_{k}(y) = … CCC … with y[0, qlp_{k} − 1] sitting directly above C). Denoting by A_{k} the zero coordinate partition of \overline{Y}_{k}, we have

H(A_{k}^{lp_{k}}|μ_{C}) = \frac{1}{qlp_{k}} ∑_{i=1}^{qlp_{k}} log #B(i, lp_{k}, C) .

The above approximately equals (the error estimate is given with the ± sign)

\frac{1}{qlp_{k}} ∑_{i=1}^{ql} p_{k}[log #B(ip_{k}, lp_{k}, C) ± p_{k} log λ]

=[\frac{1}{ql} ∑_{i=1}^{ql} ∑_{j=0}^{l−1} log α_{k}(R^{(i+j)})] ± p_{k} log λ , with i + j regarded modulo q_{l},

=[\frac{1}{ql} ∑_{i=1}^{ql} l log α_{k}(R^{(i)})] ± p_{k} log λ

=[l \frac{1}{ql} ∑_{i=1}^{ql} log (⌈exp(p_{k}f_{k}(R^{(i)}))⌉)] ± p_{k} log λ

=[l \frac{1}{ql} ∑_{i=1}^{ql} log (exp(p_{k}[f_{k}(R^{(i)}) ± ϵ_{k−1}])] ± p_{k} log λ , using (g),

=[lp_{k} \frac{1}{ql} ∑_{i=1}^{ql} f_{k}(R^{(i)})] ± (p_{k} log λ + lp_{k}ϵ_{k−1}) .

Now, by (i), we can replace the above by

lp_{k}f_{k}(μ_{C}) ± (p_{k} log λ + lp_{k} (ϵ_{k−1} + ϵ_{k}) ).

Thus

\frac{1}{lp_{k}} H(A_{k}^{lp_{k}}|μ_{C}) = f_{k}(μ_{C}) ± (\frac{log λ}{l} + ϵ_{k−1} + ϵ_{k}) .

For fixed l, measures of the form μ_{C} with C = R^{(1)}R^{(2)} … R^{(ql)} are dense. It follows from continuity of H(A_{k}^{lp_{k}}|·) (Lemma 6.6) that the last formula holds for all invariant measures μ_{k} on \overline{X}_{k}. Passing with l to infinity we obtain

h(\overline{Y}_{k}|μ_{k}) = f_{k}(μ_{k}) ± (ϵ_{k−1} + ϵ_{k}) .

We now pass to the factor map π between Y and X. Consider an invariant measure μ on X. By projection to \overline{X}_{k} this measure produces a sequence of measures μ_{k} supported by X_{k}. By Lemma 6.8, since f_{k}(μ) = f_{k}(μ_{k}), and by (b), we arrive at

h_{ext}^{φ}(μ) = h(μ) + h(Y |μ) = h(μ) + lim_{k} f_{k}(μ) = E_{A}(μ)

which concludes the proof of the Sex Entropy Theorem in Case Z. □

7. Proof of the Sex Entropy Theorem: the general case

To complete the proof of the Sex Entropy Theorem, we will lift general systems to zero dimensional systems by principal extensions.

Definition 7.1. [Le] An extension φ: (X′, T′) → (X, T) is a principal extension if h(ν) = h(φν) for every ν ∈ K′ = K(X′, T′).

Lemma 7.2. Let φ: (Y, S) → (X, T) be an extension of (X, T) and let φ′: (X′, T′) → (X, T) be a principal extension of (X, T). Then the fiber product (Z, S′) of these extensions, defined as

Z = {(y, x′): φ(y) = φ′(x′)}

with S′ = (S × T′)|_{Z}, is a principal extension of (Y, S).

Proof. Z projects naturally to both Y and X′ so that the following diagram commutes:

Z
↙ ↘
Y X′
↘ ↙
X

In this proof it will be convenient to use a classical notation for measure-theoretic entropies and conditional entropies with respect to given σ-fields [Pa]. Let ν denote an invariant measure supported by Z, and let B_{Z}, B_{Y}, B_{X′} and B_{X} denote the Borel σ-fields of the respective spaces, lifted to Z. Then, because B_{X′} ≽ B_{X}, we have

h_{ν}(B_{Y} |B_{X′}) ≤ h_{ν}(B_{Y} |B_{X}).

On the other hand, since X′ is a principal extension of X,

h_{ν}(B_{Y} |B_{X′}) = h_{ν}(B_{Y} ∨ B_{X′}|B_{X}) − h_{ν}(B_{X′}|B_{X}) ≥ h_{ν}(B_{Y} |B_{X}) − 0 .

So, h_{ν}(B_{Y} |B_{X′}) = h_{ν}(B_{Y} |B_{X}), and hence

h_{ν}(B_{Z}) = h_{ν}(B_{Y} ∨ B_{X′}) = h_{ν}(B_{Y} |B_{X′}) + h_{ν}(B_{X′})
= h_{ν}(B_{Y} |B_{X}) + h_{ν}(B_{X}) = h_{ν}(B_{Y} ∨ B_{X}) = h_{ν}(B_{Y}) .
□

Remarks 7.3. Asymptotic h-expansiveness was introduced by Misiurewicz [M1] as a sufficient condition for the existence of a measure of maximal entropy. For background, see [M1, M2, BFF, DGS] and their references. For simplicity, we appeal to the following characterization [D2, Theorem A.1]: a zero dimensional dynamical system (X, T) is asymptotically h-expansive if and only if it is topologically conjugate to a subsystem of a finite entropy countable product of subshifts. It is easy to see that the natural entropy structure of such a product converges uniformly, and this uniform convergence is inherited by systems conjugate to subsystems.

The next theorem is a special case of Ledrappier’s result that principal extensions respect the topological conditional entropy defined by Misiurewicz [M2].

Theorem 7.4. [Le] Let (X′, T′) be a principal extension of (X, T). Then (X′, T′) is asymptotically h-expansive if and only if (X, T) is asymptotically h-expansive.

For the proof of the Sex Entropy Theorem, we only use Theorem 7.4 to know in the proof of Theorem 7.5 below that a zero dimensional principal extension of an asymptotically h-expansive system must be asymptotically h-expansive. Theorem 7.5 shows that passing to principal extensions does not restrict much the variety of symbolic extensions.

Theorem 7.5. Let (X, T) be an arbitrary topological dynamical system with finite topological entropy. Let φ: (X′, T′) → (X, T) be a zero dimensional principal extension. If π: (Y, S) → (X, T) is a symbolic extension, then (Y, S) has a symbolic principal extension (Y′, S) which is an extension of (X′, T′).

Proof. The fiber product (Z, S′) of Lemma 7.2 is a principal extension of the symbolic (and hence zero dimensional and asymptotically h-expansive) system (Y, S). By Theorem 7.4, (Z, S′) is asymptotically h-expansive itself. Z is zero dimensional, as a subspace of the zero dimensional product Y ×X′, so it follows from Remarks 7.3 that H converges uniformly, where H is defined by 5.2(I) from some refining sequence of clopen partitions. It then follows from Proposition 3.2 that EH = h, so by Case Z of the Sex Entropy Theorem, (Z, S′) admits a principal symbolic extension, which then becomes both an extension of (X′, T′) and a principal extension of (Y, S), as required. □

Let P_{k} = {P_{k,i}} be a refining sequence of finite small-boundary partitions for a system (X, T) (recall Definitions 5.1). We recall a standard construction, which associates to this sequence a zero dimensional principal extension (X′, T′) of (X, T) (and which reduces to the construction described in Notation 6.2 when the P_{k,i} are clopen). Given x in X and k ∈ ℕ, define a sequence x^{k} = (x_{n}^{k})_{n∈ℤ} by setting x_{n}^{k} = i ⟺ T^{n}(x) ∈ P_{k,i}. Let X′_{k} be the closure of {x^{k}: x ∈ X}. The rule x^{k+1} ↦ x^{k} determines a factor map of subshifts (X′_{k+1}, S) → (X′_{k}, S). These maps, used as bonding maps, define (X′, T′) as an inverse limit system. The map π: (X′, T′) → (X, T) is the equivariant map which sends the subset of (X′, T′) with zero coordinate i in the coordinate for X′_{k} onto the closure of P_{k,i}. Any point of X with more than one preimage under π must lie in the boundary of some set T^{j}P_{i,k}. The union of all such boundaries has measure zero for every μ in K(X, T), and therefore the inverse image of this union has measure zero for every μ in K(X′, T′). It follows that the map π induces an affine homeomorphism π: K(X′, T′) → K(X, T) which for every μ in K(X′, T′) is a measure theoretic isomorphism between the measure preserving systems (X′, μ, S) and (X, πμ, T), and in particular respects measure theoretic entropy, so π is a principal extension. Finally, the entropy structure H on K(X, T) given by h_{k}(μ) = h(T, μ, P_{n}) is simply the renaming by π of the standard entropy structure of the inverse limit system K(X′, T′), which we will call H′.

With this in hand, it is now easy to finish the proof of Theorem 5.5 in the case that the entropy structure H on (X, T) is given from a sequence P_{k} as above.

Proof of Theorem 5.5 in Case I of H.

First, if E is a bounded affine superenvelope of H, then E′ := E ∘π is a bounded affine superenvelope of H′, so by the Case Z of Theorem 5.5 proved in the last section there is a symbolic extension φ of (X, T) such that h_{ext}^{φ} = E′, and then π ∘ φ is a symbolic extension of (X, T) such that h_{ext}^{π∘φ} = E. Conversely, if φ: (Y, S) → (X, T) is a symbolic extension of (X, T), then by Theorem 7.5 there is a symbolic extension ψ: (Y′, S) → (X′, T′) such that h_{ext}^{φ} = h_{ext}^{ψ} ∘ π. Because h_{ext}^{ψ} is a bounded affine superenvelope of H′, it follows that h_{ext}^{φ} is a bounded affine superenvelope of H. □

Existence of principal extensions

Not every system has a refining sequence of small boundary partitions; for example, an interval of fixed points is an obstruction. In the next theorem, we note that many do. The theorem is essentially a collection of results of Kulesza, Lindenstrauss and Weiss.

Theorem 7.6. Each of the following conditions is sufficient to guarantee that a system (X, T) has a refining sequence of small boundary partitions.

(1) X is zero dimensional.

(2) X is finite dimensional and the periodic point set of T is zero dimensional.

(3) (X, T) has an infinite minimal factor and h_{top}(T) < ∞.

Remark 7.7. It is the last condition (3) on which we will rely to construct an entropy structure for a general system.

Proof. (1) This is trivial.

(2) This is essentially contained in [Ku], as described in [BFF, App. B]).

(3) In this case the existence of the small boundary partitions follows from two deep results of Lindenstrauss and Weiss involving the mean dimension (an idea suggested by Gromov). First, every finite entropy topological dynamical system has zero mean dimension ([LiWe, Theorem 4.2] and the discussion above it); and second, every zero mean dimension dynamical system admitting a nonperiodic minimal factor has the so called small boundary property [Li, Theorem 6.2], which implies the existence of a basis of the topology consisting of sets whose boundaries have measure zero for every invariant measure. With these results it is easy to construct the refining sequence of partitions with small boundaries. □

We get as a corollary a very general statement.

Proposition 7.8. Every finite entropy system (X, T) has a zero dimensional principal extension.

Proof. Let (Z, R) be an infinite, zero entropy minimal system. Then (X ×Z, T ×R) is a principal extension of (X, T). By Fact 7.6(3), (X × Z, T × R) has a zero dimensional principal extension, which by composition is a principal extension of (X, T). □

In the special case that (X, T) is asymptotically h-expansive, there is a more direct construction of a zero dimensional principal extension [BFF, A.2 and 6.7].

Proof of Theorem 5.5 in Case II of H.

We isolate the main argument of the proof as the next lemma.

Lemma 7.9. Given a system (X, T), let (Z, R) be an aperiodic minimal system with an R-invariant Borel probability measure λ such that h(R, λ) = 0 = h_{top}(R). Set (X′, T′) = (X × Z, T × R) and let π : X′ → X denote the projection on the first axis. Let (α_{k}) be a refining sequence of partitions on X′ with small boundaries for the system (X′, T′) and let H′ = (h′_{k}) be the corresponding entropy structure on K(X′, T′), defined for T′ as in 5.2(I). Let H′′ = (h′′_{k}) be the sequence of functions on K(X′, T′) defined by h′′_{k}(μ′) = h′_{k}(μ × λ), where μ = πμ′.

Then the following are equivalent.

(1) E is a bounded superenvelope for H′′.

(2) E is a bounded superenvelope for H′.

Proof. Let h denote the entropy function on K(X′, T′). Obviously, h is the limit of both H′ and H′′.

(1) ⟹ (2) Let E be a bounded superenvelope of H′′. There is no inequality granted between h′_{k} and h′′_{k}. However, given k, we do have E − h′_{k} ≥ E − h = lim_{n} E − h′′_{n}, which is nonnegative because E is a superenvelope of H′′. It remains to show E − h′_{k} is u.s.c. Fix μ′ ∈ K(X′, T′) and consider the defect

lim sup_{μ′_{n}→μ′}(E − h′_{k})(μ′_{n}) − (E − h′_{k})(μ′).

Let D be the compact countable set {μ′, μ′_{n}, μ × λ, μ_{n} × λ: n ∈ ℕ}, where (μ′_{n}) is a sequence which realizes the above limsup. We will restrict our considerations to D. We have

E − h′_{k} = (E − h′′_{ℓ}) + (h′′_{ℓ} − h′_{k}),

for any ℓ > k. Since the first bracket (E − h′′_{ℓ}) is u.s.c., it suffices to estimate the defect of upper semicontinuity of the second bracket (restricted to D) at μ′, for which we can choose ℓ as large as we need. Because D is countable, we can find a partition of X, say β, with boundaries of measure zero for all measures arising as projections of measures from D. Also, because α_{k} has small boundaries, by choosing this partition β fine enough and by choosing a partition γ of Z fine enough, we can approximate the partition α_{k} by a coarsening β_{1} of β × γ such that for all measures in D,

|h(·, β_{1}) − h(·, α_{k})| < \frac{ϵ}{2} .

On account of the null boundaries and coarsening, the function h(·, β ×γ)−h(·, β_{1}) is u.s.c. on D. Further, we can approximate β × γ by a coarsening β_{2} of a partition α_{ℓ} (for some large ℓ) so that

|h(·, β_{2}) − h(·, β × γ)| < \frac{ϵ}{2}

on D. Also, h(·, α_{ℓ}) − h(·, β_{2}) is u.s.c. (this holds not only on D, but even on K). Now observe that h(ν′, β × γ) = h(ν × λ, β × γ), because the partition is a product partition and h_{top}(R) = 0. Thus the function h′′_{ℓ} − h′_{k} at a point ν′ ∈ K(X′, T′) can be written as

[h(ν × λ, α_{ℓ}) − h(ν × λ, β_{2})] + [h(ν × λ, β_{2}) − h(ν × λ, β × γ)] +
[h(ν′, β × γ) − h(ν′, β_{1})] + [h(ν′, β_{1}) − h(ν′, α_{k})] .

Only the second and fourth brackets contribute to the defect of upper semicontinuity on D, and their joint contribution is less than ϵ.

(2) ⟹ (1) Let E be a superenvelope of H′. The argument for nonnegativity of E − h′′_{k} is essentially the same as for the first implication. The argument for uppersemicontinuity is also essentially the same, after reversing the roles of ν′ and ν × λ:

E − h′′_{k} = (E − h′_{ℓ}) + (h′_{ℓ} − h′′_{k})

and

(h′_{ℓ} − h′′_{k})(ν′) = [h(ν′, α_{ℓ}) − h(ν′, β_{2})] + [h(ν′, β_{2}) − h(ν′, β × γ)] +
[h(ν × λ, β × γ) − h(ν × λ, β_{1})] + [h(ν × λ, β_{1}) − h(ν × λ, α_{k})].
□

We now prove the implications of the Sex Entropy Theorem for Case II. We continue the notation of Lemma 7.9.

Proof of Theorem 5.5 (1) ⟹ (2) in Case II of H.

Suppose E is a bounded affine superenvelope of H. By Lemma 7.9, the pullback E′′ of E to K(X′, T′) is a bounded affine superenvelope of H′. By the Sex Entropy Theorem in Case I, there is a symbolic extension φ of (X × Z, T × R) such that h_{ext}^{φ} = E′′. Then h_{ext}^{π∘φ} = E.

Proof of Theorem 5.5 (2) ⟹ (1) in Case II of H.

Suppose φ: (Y, S) → (X, T) is a symbolic extension and E = h_{ext}^{φ}. By Theorem 7.5, let ψ: (Y′, S) → (X′, T′) be a symbolic extension such that (Y′, S) is a principal extension of (Y, S), so E = h_{ext}^{φ} = h_{ext}^{π∘ψ} = (h_{ext}^{ψ})^{[K]} where K = K(X, T). Because h_{ext}^{ψ} is a bounded affine superenvelope of H′, by Lemma 7.9 it is also a bounded affine superenvelope of H′′, and therefore each h_{ext}^{ψ} − h′′_{k} is u.s.c. It then follows from Remark 2.6 that (h_{ext}^{ψ} − h′′_{k})^{[K]} is u.s.c. Because h′′_{k} is constant and equal to h_{k} on fibers of π, we have (h_{ext}^{ψ} − h′′_{k})^{[K]} = (h_{ext}^{ψ})^{[K]} − h_{k} = E − h_{k}, so E is a superenvelope of H. As we have seen, it follows from Proposition 2.7 that E = h_{ext}^{π∘ψ} is affine.

□

Remark 7.10. With fiber product constructions, we could have avoided the use of Lemma 7.9 above for the direction (2) ⟹ (1).

8. Consequences for sex entropy

In this section we collect various (at this point easy) corollaries of our work for sex entropy. Recall our conventions: the infimum of the empty set is +∞, and the only unbounded superenvelope is the constant function +∞.

Theorem 8.1. The sex entropy function h_{sex}(T, ·) is the upper semicontinuous concave function EH which is the smallest superenvelope of the entropy structure H. Moreover, there is a variational principle for sex entropy:

h_{sex}(T) = SUP(h_{sex}(T, ·)) := sup_{μ} inf_{φ}{h_{ext}^{φ}(μ): φ is a symbolic extension of T } .

Proof. It suffices to consider the case in which h_{sex}(T) is finite (equivalently, H has a bounded superenvelope). The first claim is a consequence of the Sex Entropy Theorem and the fact (Proposition 4.3) that EH is the infimum of the affine superenvelopes of H. The variational principle then follows from the variational statement in Proposition 4.3. □

Remark 8.2. There is of course no variational principle for residual entropy. Theorem 8.1 implies

h_{res}(T) ≤ SUP(h_{res}(T, ·)) := sup_{μ} inf_{φ}{h_{ext}^{φ}(μ)−h(μ): φ is a symbolic extension of T }

but it is easy to provide examples in which the inequality is strict. (For example, let (X, T) be the disjoint union of a subshift (X_{1}, S) and a system (X_{2}, R) such that h_{top}(S) ≥ h_{sex}(R) > h_{top}(R).) This is an example of “sex entropy” being notationally more convenient than “residual entropy”.

The characterization of sex entropy as EH has considerable content on account of the functional analysis results in sections 2, 3 and 4. The results there for a sequence F of u.s.c. functions with u.s.c. differences apply to any entropy structure H = (h_{k})_{k}. In particular, the characterizations of superenvelopes by continuous covers (Theorem 2.18) and by the inductive construction of Theorem 3.3 apply. The transfiniteness of the last construction explains for us some of the subtlety of residual entropy.

We apply the mix of functional analysis and the Sex Entropy Theorem in the next theorem, a collection of attainability results.

Theorem 8.3 (Attainability Results). Suppose (X, T) is a system with entropy structure H and space of invariant Borel probabilities K(X, T), and (X, T) has some symbolic extension (i.e. EH is bounded).

(1) There is a symbolic extension φ such that h_{ext}^{φ} = h_{sex}^{T} if and only if EH is affine. If the ergodic measures are a closed subset of K(X, T), then EH is affine.

(2) There is a symbolic extension (Y, S) of (X, T) such that h_{top}(S) = h_{sex}(T) ⟺ there exists an affine superenvelope E_{A} of H with SUP(E_{A}) = SUP(EH).

(3) The sex entropy function h_{sex}(T, ·) on K(X, T) achieves its maximum on the closure of the ergodic measures.

Proof. The characterizations in (1) and (2) follow from the Sex Entropy Theorem 5.5 and the characterization of h_{sex}(T, ·) as EH. The claim in (1) involving ergodic measures follows from the Bauer simplex statement in Theorem 4.6 with F replaced by H. The claim (3) follows from h_{sex}(T, ·) = EH and Theorem 4.9. □

Similarly, the results of Sections 2 and 3 for estimating or computing EF apply to EH. Moreover, all abstract candidates F actually occur as entropy structures:

Theorem 8.4. Entropy Structure Realization Theorem [DS2, Theorem 3] Let K be a Choquet simplex and let F = (f_{k})_{k∈ℕ} be a nondecreasing sequence of nonnegative affine u.s.c. functions on K such that each f_{k} − f_{k−1} is u.s.c. Then there exist a zero dimensional dynamical system (X, T) in the form of an inverse limit of subshifts (X, T) = \overleftarrow{lim}_{k}(X_{k}, S), and an affine homeomorphism a: K(X, T) → K, such that for every invariant measure μ on X and every k,

(f_{k} ∘ a)(μ) = h(μ_{k}).

Consequently, various entropy examples are immediate consequences of analytic examples, the Sex Entropy Theorem and the Entropy Structure Realization Theorem. Here are some instances of this.

Theorem 8.5 (Nonattainment examples).

(1) There is a system (X, T) such that for every symbolic extension φ: (Y, S) → (X, T), h_{top}(S) > h_{sex}(T). Moreover, this can happen even if T satisfies h_{sex} = h̃.

(2) There is a system (X, T) such that h_{sex}(T, μ) equals 1 for ergodic μ, but SUP(h_{sex}(T, ·)) > 1.

Proof. By the Entropy Structure Realization Theorem, there are systems with entropy structures H matching the sequences F in Example 4.7 and Example 4.8. Now apply the Sex Entropy Theorem.

□

Similarly the examples 2.19, 2.21 and 2.20 extend to entropy structures on Bauer simplices (see Remark 4.5). We take two relevant superenvelope examples to the level of concrete dynamical systems at the end of this section.

First, though, we want to exhibit a theorem summarizing equivalent conditions for asymptotic h-expansiveness, which is a significant condition for smooth as well as topological dynamics. (A C^{∞} system must be asymptotically h-expansive, by Buzzi [Bu], following Yomdin – see [BFF, Theorem 7.8] – but for r < ∞, a C^{r} system need not be asymptotically h-expansive, because upper semicontinuity of the entropy function can fail [M1, N].) The main part of this theorem, the equivalence below of (1) and (2), is known; but we will see how the asymptotically h-expansive systems appear very naturally as a distinguished class in the superenvelope/sex entropy setting.

Theorem 8.6 (Asymptotic h-expansiveness). The following are equivalent for a system (X, T) with entropy structure H.

(1) (X, T) is asymptotically h-expansive.

(2) (X, T) has a symbolic extension which is a principal extension.

(3) EH = h.

(4) h_{k} converges uniformly to h.

Proof. (2) ⟺ (3) follows from the Sex Entropy Theorem (because h is affine). (3) ⟺ (4) follows from Proposition 3.2 with F = H.

(1) ⟹ (2) was proved independently in [D2] (in the zero dimensional case) and [BFF] (in general).

(2) ⟹ (1) follows immediately from Theorem 7.4. □

Remark 8.7. The proof above for equivalence of (1) and (2) left the framework of the current paper. In [D3], a generalized approach to entropy structure gives a very natural proof for the equivalence of (1) and (4), which puts the proof of Theorem 8.6 entirely into the entropy structure setting.

Remark 8.8. Kifer and Weiss [KiWe] have a “relative” theorem analogous to (1) ⟹ (2) above. They introduced asymptotic entropy expansive random transformations, and proved a generator theorem allowing their representation by random symbolic subshifts.

Remark 8.9. Let H be an entropy structure. We have from Theorem 2.18 that SUP(EH)−SUP(h) = inf_{G} SUP(ΣG)−h_{top}(T), where G denotes a continuous cover of H. Because h_{res}(T) = h_{sex}(T) − h_{top}(T) = SUP(EH) − h_{top}(T), we recover for general X the characterization of h_{res}(X, T) given for the zero dimensional case in [D2]: h_{res}(T) = inf_{G} SUP(ΣG) − h_{top}(T). Note, Theorem 2.18 shows that this infimum is always achieved by some cover G, so the continuous covers alone cannot be adequate to detect the nonattainment phenomenon of Theorem 8.5(1).

We now turn to the promised examples. Other constructions of this kind were presented in [D2], achieving all values for the topological, residual and topological conditional entropies consistent with the known constraints. Quite different methods in [BFF] produced another rich family of examples.

Example 8.10. This is a very simple example with nonzero residual entropy. The strategy is simply to construct a system with H = F as in Example 2.19.

Let (X_{0}, S) denote a strictly ergodic {0-1}-subshift with invariant measure μ_{0} and entropy 1. Let B_{k} (k ≥ 2) be a sequence of blocks occurring in X_{0} of lengths increasing with k and such that μ_{B_{k}} → μ_{0}. Let X denote the set of all {0-1}-valued matrices x = (x_{k,n})_{k∈ℕ,n∈ℤ} satisfying the following conditions.

(1) The first row x_{1} of x either belongs to X_{0} or has the form x_{B_{k}} = … B_{k}B_{k}B_{k} … for some k ≥ 2. By taking closure we must also admit that x_{1} may have the form y_{1}(−∞, m]y_{2}[m + 1, ∞) for some y_{1}, y_{2} ∈ X_{0}, m ∈ ℤ.

(2) If the first row is x_{B_{k}}, then the kth row of x is an element of X_{0}.

(3) All other rows are filled with zeros. The set of matrices so constructed is closed and invariant under the horizontal shift Sx_{k,n} = x_{k,n+1}. The structure of invariant measures is as follows: there is one measure μ^{(1)} supported by matrices having nonperiodic first row and all other rows filled with zeros; this measure is isomorphic to μ_{0}, its entropy is 1 and h_{k}(μ^{(1)}) = 1 for k ≥ 1. Moreover, for each k > 1, there are finitely many measures μ^{(k,i)} supported by matrices having nonzero kth row and periodic first row. Again, each of these measures has entropy 1, with h_{n}(μ^{(k,i)}) = 1 for n ≥ k and h_{n}(μ^{(k,i)}) = 0 for n < k. All measures μ^{(k,i)} accumulate at μ^{(1)}. The entropy function h is 1 on all measures, so h_{top}(T) = 1, while EH = 1 + 1_{{μ^{(1)}}}^{aff}, hence h_{sex}(X, S) = 2.

Example 8.11. In this example, h_{top}(S) > h_{sex}(T) for every symbolic extension S of T. (Moreover, EH = h̃, in particular h_{res}(T) = 0.) The idea is to obtain a system for which the entropy sequence behaves as in the abstract Example 4.7.

Let X_{0} and B_{k} be as in the previous example. Let C_{k} (k ≥ 2) be a block of the following structure:

C_{k} = B_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B_{2}B_{3} . . . . . . . . . . . . B_{3} ⋯ B_{k} … B_{k},

where the repetitions of B_{i+1} (numerous even for i = k) occupy roughly 2^{−i} of the length of C_{k}, and the precision of these proportions improves with k (the number of repetitions for each i increases with k). It is now seen that the measures μ_{C_{k}} carried by the periodic orbits of x_{C_{k}} = … C_{k}C_{k}C_{k} … converge weakly* to ∑_{k=1}^{∞} 2^{−k}μ_{B_{k+1}}. Let X be the set of all {0-1}-valued matrices x = (x_{k,n})_{k∈ℕ,n∈ℤ} satisfying the following conditions.

(1) The first row x_{1} of x either belongs to X_{0} or it has the form x_{B_{k}} for some k ∈ ℕ, or it is x_{C_{k}} for some k ≥ 2. By taking closure we must additionaly admit the forms y_{1}(−∞, m]y_{2}[m + 1, ∞) with y_{1}, y_{2} ∈ X_{0}, or x_{B_{k}}(−∞, m]x_{B_{k+1}}[m + 1, ∞) and also y_{1}(−∞, m]x_{B_{2}}[m + 1, ∞).

(2) If the first row is x_{C_{k}}, then the kth row of x is an element of X_{0}.

(3) All other rows are filled with zeros.

The structure of invariant measures is now the following: the measure μ^{(1)} is as in the previous example, and h_{1} is the (affine extension of) the characteristic function at μ^{(1)}. This measure is approached by periodic measures μ^{(k)} supported by matrices with x_{B_{k}} in the first row (this time there is only one such measure for each k ≥ 2, so the index i can be dropped; moreover, these measures now have entropy zero). In addition, for each k ≥ 2, we have finitely many measures ν^{(k,j)} supported by matrices with x_{C_{k}} in the first row and a nonperiodic kth row. Then h_{k} is the (affine extension of) characteristic function of the set of measures ν^{(k,j)}. With increasing k, these latest measures approach the combination ∑_{k=1}^{∞} 2^{−k}μ^{(k+1)}. The behaviour of H of Example 4.7 is hence copied.

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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  Home page:  www.math.umd.edu/∼mmb

Institute of Mathematics, Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

   E-mail address: downarim.pwr.wroc.pl  Home page:  www.im.pwr.wroc.pl/∼downar