Existence of a measurable saturated compensation function between subshifts and its applications

TL;DR

This paper proves the existence of a measurable saturated compensation function between subshifts and applies it to determine Hausdorff dimensions and measures of full dimension for invariant sets under expanding nonconformal maps, extending previous results.

Contribution

It introduces a measurable saturated compensation function for subshifts and extends dimension formulas to more general symbolic representations.

Findings

Existence of a bounded Borel measurable saturated compensation function.

Derived a formula for Hausdorff dimension without the shift of finite type condition.

Characterized invariant measures of full dimension as ergodic equilibrium states.

Abstract

We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find 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. These problems were studied in [19] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. We extend the results by presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift without the condition and characterizing the invariant ergodic measures of full dimension as the ergodic equilibrium states of a constant multiple of a measurable compensation function. For a compact invariant set whose symbolic representation is a topologically mixing shift of finite type, we study uniqueness and the properties for the unique invariant ergodic measure of full dimension by using a measurable compensation function. Our positive results narrow the possibility of where an example having more than one measure of full dimension can be found.