Abstract
We construct a self-affine sponge in mathbb {R}^3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space.
Highlights
Self-affine sponges: a dimension gap the Hausdorff dimension of that set
If the Hausdorff dimension of an invariant measure is strictly less than the Hausdorff dimension of the entire fractal, the set of typical points for the measure is much smaller than the set of atypical points, and the dynamics of “most” points on the fractal are not captured by the measure
McCluskey and Manning [33] showed that “most” two-dimensional Axiom A diffeomorphisms have basic sets whose Hausdorff dimension is strictly greater than their dynamical dimension, and in particular there are no invariant measures of full dimension
Summary
Self-affine sponges: a dimension gap the Hausdorff dimension of that set. An ergodic invariant measure with this property can be thought of as capturing the “typical” dynamics of points on the fractal. In this paper we will prove that the answer to Question 1.2 is negative by constructing a piecewise affine expanding repeller topologically conjugate to the full shift whose Hausdorff dimension is strictly greater than its dynamical dimension This expanding repeller will belong to a class of sets that we call “self-affine sponges” (not all of which are expanding repellers), and we develop tools for calculating the Hausdorff and dynamical dimensions of self-affine sponges more generally. We write the Hausdorff dimension of a self-affine sponge as the supremum of the Hausdorff dimensions of certain nice non-invariant measures that we call “pseudo-Bernoulli” measures (see Definition 2.10), which are relatively homogeneous with respect to space, but whose behavior with respect to length scale varies in a periodic way The dimension of these measures turns out to be calculable via an appropriate analogue of the Ledrappier–Young formula, which is how we show that it is sometimes larger than the dimension of any invariant measure
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