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

Magdalena Foryś-Krawiec; Jiří Kupka; Piotr Oprocha; Xueting Tian

doi:10.3934/dcds.2020317

Article Contents

2021, Volume 41, Issue 3: 1271-1296. Doi: 10.3934/dcds.2020317

This issue Previous Article On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations Next Article Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium

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

1.
National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic
2.
National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic, – and –, AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
3.
School of Mathematical Science, Fudan University, Shanghai 200433, China

^* Corresponding author: oprocha@agh.edu.pl
^* Corresponding author: oprocha@agh.edu.pl

Received: October 31, 2019

Revised: April 30, 2020

Early access: August 2020

Published: March 2021

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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.

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Mathematics Subject Classification: 37B20, 37B40.

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