Two results on entropy, chaos and independence in symbolic dynamics

Fryderyk  Falniowski; Marcin  Kulczycki; Dominik Kwietniak; Jian Li

doi:10.3934/dcdsb.2015.20.3487

Article Contents

2015, Volume 20, Issue 10: 3487-3505. Doi: 10.3934/dcdsb.2015.20.3487

This issue Previous Article A note on specification for iterated function systems Next Article Entropy determination based on the ordinal structure of a dynamical system

Two results on entropy, chaos and independence in symbolic dynamics

1.
Department of Mathematics, Cracow University of Economics, ul. Rakowicka 27, 31-510 Kraków
2.
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, ul. Lojasiewicza 6, 30-348 Kraków
3.
Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, ul. Łojasiewicza 6, 30-348 Kraków
4.
Department of Mathematics, Shantou University, Shantou, Guangdong 515063

Received: December 2014

Revised: March 2015

Published: December 2015

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Abstract

We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics:
1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces.
2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair.
Our proofs are new and yield conclusions stronger than what was known before.

Keywords:

Mathematics Subject Classification: 37B40, 37B10.

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