Sensitive dependence on initial conditions and chaotic group actions

Author:
Fabrizio Polo

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2815-2826

MSC (2010):
Primary 28D05, 28D15, 37A05, 37B05; Secondary 22B99

DOI:
https://doi.org/10.1090/S0002-9939-10-10286-X

Published electronically:
April 14, 2010

MathSciNet review:
2644895

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Abstract | References | Similar Articles | Additional Information

Abstract: A continuous action of a group on a compact metric space has *sensitive dependence on initial conditions* if there is a number such that for any open set we can find such that has diameter greater than We prove that if a countable acts transitively on a compact metric space, preserving a probability measure of full support, then the system either is minimal and equicontinuous or has sensitive dependence on initial conditions. Assuming ergodicity, we get the same conclusion without countability. These theorems extend the invertible case of a theorem of Glasner and Weiss. We prove that when a finitely generated, solvable group acts transitively and certain cyclic subactions have dense sets of minimal points, the system has sensitive dependence on initial conditions. Additionally, we show how to construct examples of non-compact monothetic groups and transitive, non-minimal, almost equicontinuous, recurrent -actions.

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Additional Information

**Fabrizio Polo**

Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210

Email:
polof@math.osu.edu

DOI:
https://doi.org/10.1090/S0002-9939-10-10286-X

Received by editor(s):
July 14, 2009

Received by editor(s) in revised form:
November 12, 2009, and November 13, 2009

Published electronically:
April 14, 2010

Communicated by:
Bryna Kra

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.