The Asymptotically Additive Topological Pressure: Variational Principle For Non Compact and Intersection of Irregular Sets

TL;DR

This paper establishes a variational principle for asymptotically additive topological pressure in certain non-compact dynamical systems, extending classical results and analyzing irregular sets with full pressure.

Contribution

It introduces a variational principle for non-compact systems with asymptotically additive pressure using g-almost product and uniform separation properties.

Findings

Proves the variational principle for non-compact systems.

Shows irregular sets carry full asymptotically additive topological pressure.

Applicable to systems like shifts, repellers, and hyperbolic diffeomorphisms.

Abstract

Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space and $f:X\rightarrow X$ is a continuous map. Using the concepts of \textit{g-almost product property} and \textit{uniform separation property} introduced by Pfister and Sullivan in \cite{Pfister2007}, we give a variational principle for certain non-compact with relation the asymptotically additive topological pressure. We also study the set of points that are irregular for an collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure. These results are suitable for systems such as mixing shifts of finite type, $\beta$-shifts, repellers and uniformly hyperbolic diffeomorphisms.