Topological pressure of simultaneous level sets

TL;DR

This paper develops a thermodynamic approach to analyze the topological pressure of level sets in dynamical systems, extending multifractal analysis to broader classes of potentials and non-uniformly expanding maps.

Contribution

It introduces a method to study pressure on level sets using equilibrium states, extending previous results from smooth to less regular systems and potentials.

Findings

Extended multifractal results to $C^1$ systems.

Analyzed pressure for non-uniformly expanding maps.

Covered coarse spectra and general limiting behaviors.

Abstract

Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets. Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace. This technique allows us to extend a number of previous multifractal results from the $C^{1+\epsilon}$ case to the $C^1$ case. We consider ergodic ratios $S_n \phi/S_n \psi$ where the function $\psi$ need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.