Variational principles for topological pressures on subsets

TL;DR

This paper develops new variational principles linking topological and measure-theoretic pressures, extending existing theories and providing tools to analyze pressures on subsets in dynamical systems.

Contribution

It extends Feng-Huang's variational principle to packing pressure and introduces new principles for Pesin-Pitskel and packing pressures, advancing the understanding of pressures in dynamical systems.

Findings

Established variational principles for packing and Pesin-Pitskel pressures.

Proved equality of Katok pressures to entropy plus potential integral.

Derived Billingsley type theorem relating packing pressure to local measure pressures.

Abstract

In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng- Huang's variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin-Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.