The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result

TL;DR

This paper constructs a 3D self-affine sponge demonstrating a strict difference between its Hausdorff and dynamical dimensions, resolving a long-standing open problem and extending dimension theory to higher dimensions.

Contribution

It introduces a new class of self-affine sponges in three dimensions with a proven dimension gap, advancing the understanding of dimension theory in dynamical systems.

Findings

Existence of a 3D self-affine sponge with a dimension gap

Explicit computation of Hausdorff and dynamical dimensions for a broad class of sponges

Continuity of dimensions with respect to the defining iterated function system

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.