Effective uniqueness of Parry measure and exceptional sets in ergodic theory

TL;DR

This paper provides an effective version of the uniqueness of the Parry measure in hyperbolic systems and estimates the Hausdorff dimension of exceptional sets in ergodic theory.

Contribution

It introduces an effective approach to the uniqueness of Parry measure and derives bounds on the Hausdorff dimension of exceptional sets.

Findings

Effective bounds for the uniqueness of Parry measure

Upper estimates for Hausdorff dimension of exceptional sets

Application to hyperbolic dynamical systems

Abstract

It is known that hyperbolic dynamical systems admit a unique invariant probability measure with maximal entropy. We prove an effective version of this statement and use it to estimate an upper bound for Hausdorff dimension of exceptional sets arising from dynamics.