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

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.

Cited by in corpus (10)

• Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension
• Badly approximable points on self-affine sponges and the lower Assouad dimension
• The $L^q$ spectrum of self-affine measures on sponges
• Triangular Gatzouras-Lalley-type planar carpets with overlaps
• How long is the Chaos Game?
• The Assouad dimension of self-affine measures on sponges
• Totally ergodic generalised matrix equilibrium states have the Bernoulli property
• Assouad spectrum of Gatzouras-Lalley carpets
• $L^q$-spectra of box-like graph-directed self-affine measures: closed forms, with rotation
• On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets