Dimensions of compact invariant sets of some expanding maps

TL;DR

This paper investigates the Hausdorff dimension and measures of full dimension for invariant sets of expanding nonconformal maps on the torus, extending previous results on Sierpinski carpets using shift of finite type models.

Contribution

It extends existing results by applying compensation functions to analyze general Sierpinski carpets and establishes conditions for the uniqueness of measures of full Hausdorff dimension.

Findings

Identifies conditions for unique measures of full Hausdorff dimension.

Extends analysis to a broader class of Sierpinski carpets.

Uses shift of finite type to model complex invariant sets.

Abstract

We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpinski carpet represented by a shift of finite type. We give some conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension, and study the properties of the unique measure.