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Combinatorial independence and naive entropy

Part of: Extremal combinatorics Ergodic theory Topological dynamics

Published online by Cambridge University Press:  15 May 2020

HANFENG LI  and
ZHEN RONG
HANFENG LI
Affiliation:
Center of Mathematics, Chongqing University, Chongqing401331, China Department of Mathematics, SUNY at Buffalo, Buffalo, NY14260-2900, USA email hfli@math.buffalo.edu
ZHEN RONG
Affiliation:
College of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot010000, China email rongzhen@cqu.edu.cn
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Abstract

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.

Keywords

naive entropycombinatorial independenceLi–Yorke chaosdistal actiontame action

MSC classification

Primary: 37B40: Topological entropy 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 05D10: Ramsey theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2136 - 2147
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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