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

Magdalena Foryś-Krawiec, Jiri Kupka, Piotr Oprocha, Xuentin Tian

Published 2019-09-18Version 1

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.