On entropy of $\Phi$-irregular and $\Phi$-level sets in maps with the shadowing property

TL;DR

This paper investigates the entropy characteristics of irregular and level sets in dynamical systems with shadowing, revealing their typicality and relation to ergodic measures and large deviations.

Contribution

It provides entropy estimates for irregular and level sets in systems with shadowing, connecting these sets to chain recurrent classes and ergodic measures.

Findings

Full entropy irregular sets are typical in systems with shadowing.

Entropy of level sets relates to entropy of ergodic measures.

Large deviations for level sets are analyzed with respect to reference measures.

Abstract

We study the properties of $\Phi$-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $\Phi$-irregular set in terms of entropy on chain recurrent classes and prove that $\Phi$-irregular sets of full entropy are typical. We also consider $\Phi$-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.