A variational principle for topological pressure for certain non-compact sets

TL;DR

This paper establishes a variational principle for topological pressure on certain non-compact sets in dynamical systems with the specification property, extending known results from entropy to pressure and applying to multifractal analysis.

Contribution

It introduces a variational principle for topological pressure on non-compact sets, generalizing previous entropy results and enabling multifractal analysis in new settings.

Findings

Proved a variational principle for topological pressure on specific non-compact sets.

Extended multifractal analysis to entropy spectrum and dimension spectrum.

Applied results to non-uniformly expanding interval maps.

Abstract

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $\varphi: X \mapsto \IR$ be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form \[ \{x \in X : \lim_{n \ra \infty} \frac{1}{n} \sum_{i = 0}^{n-1} \varphi (f^i (x)) = \alpha \}. \] Analogous results were previously known for topological entropy. As an application, we prove multifractal analysis results for the entropy spectrum of a suspension flow over a continuous map with specification and the dimension spectrum of certain non-uniformly expanding interval maps.