Nonadditive Measure-theoretic Pressure and Applications to Dimensions of an Ergodic Measure

TL;DR

This paper introduces a new framework for measure-theoretic pressure applicable to ergodic measures, linking it to entropy and dimensions, and applies it to conformal repellers to analyze their dimensional properties.

Contribution

It defines subadditive and supadditive measure-theoretic pressures without extra conditions and establishes their relation to entropy, providing new tools for dimension analysis of dynamical systems.

Findings

Ergodic measure of maximal Hausdorff dimension exists on average conformal repellers.

Constructed sets with dimensions equal to their supporting measures.

Established inverse variational principle for subadditive pressure.

Abstract

Without any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact set, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the supadditive measure-theoretic pressure which has similar formalism as the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Haudorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.