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Israel Journal of Mathematics

, Volume 183, Issue 1, pp 233–283 | Cite as

Measure-theoretical sensitivity and equicontinuity

  • Wen HuangEmail author
  • Ping Lu
  • Xiangdong Ye
Article

Abstract

For an invariant measure µ in a topological dynamics, notions of µ-sensitivity, µ-complexity and µ-equicontinuity are introduced and investigated. It turns out that µ-sensitivity defined here is equivalent to pairwise sensitivity defined by Cadre and Jacob. For an ergodic µ, µ-equicontinuity, no µ-complexity pair and non-µ-sensitivity are equivalent, which implies minimality and equicontinuity when restricted to the support. Moreover, the notion of µ-sensitive set is introduced, it is shown that a transitive system with an ergodic measure of full support has zero topological entropy if there is no uncountable µ-sensitive set, and a non-minimal transitive system with dense minimal points has infinite sequence entropy for some sequence.

For a minimal system (X, T) it is shown that (x 1, x 2) is regionally proximal if and only if, for any neighborhood U of x 2, {n∈ℤ+:T n x 1U} is a Poincaré sequence. This implies that if (x i , x i+1) is regionally proximal for i = 1, …, n − 1, then (x 1, …, x n ) is n-regionally proximal. The structure of a minimal system is determined via the cardinalities of sensitive sets for µ.

Keywords

Invariant Measure Open Neighborhood Open Cover Topological Entropy Minimal System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of China HefeiAnhuiP.R. China

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