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Brian R. Hunt

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Submitted.
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In $n$-dimensional Euclidean space, consider iterated function systems
that consist of a finite number of $C^{1+\alpha}$ invertible
contracting maps $T_i$, and associated probabilities. Such an iterated
function system corresponds to a ``baker's map'' in dimension $n+1$,
which has a natural invariant measure $\mu$ that attracts Lebesgue
almost every initial condition. This measure has an almost everywhere
constant pointwise dimension, which as a result coincides with the
Hausdorff and information dimensions of the measure. We do not assume
that the images of the support of this measure under different $T_i$
are disjoint. The main result of this paper is that if the system is
linear or conformal and depends on a number of parameters that include
arbitrary translations of the images of the contracting maps, and each
map contracts lengths by a factor less than $1/2$, then for Lebesgue
almost every set of parameters, the pointwise dimension of $\mu$ is
equal to the Lyapunov dimension of $\mu$ (which is an explicit
function of the Lyapunov exponents of $\mu$). We also discuss the
case of a system that is both nonlinear and nonconformal, and
formulate more general ``transversality'' conditions on the parameter
dependence that imply the same result.

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*Updated: June 30, 1999*