Conditional Variational Principle for Historic Set in Some Nonuniformly Hyperbolic Systems

TL;DR

This paper investigates the historic set of Birkhoff averages in certain nonuniformly hyperbolic systems, establishing a conditional variational principle that applies to various classes of diffeomorphisms, including surface diffeomorphisms.

Contribution

It introduces a conditional variational principle for the historic set in nonuniformly hyperbolic systems, extending the understanding of Birkhoff averages in these complex dynamics.

Findings

Established a variational principle for historic sets in nonuniformly hyperbolic systems

Applied results to surface diffeomorphisms and systems described by Katok

Extended the theory to classes derived from Anosov systems

Abstract

This article is devoted to the study of the historic set, which was introduced by Ruelle, of Birkhoff averges in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for historic sets. Our results can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.