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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: A continuous action of a group $ G$ on a compact metric space has sensitive dependence on initial conditions if there is a number $ \varepsilon > 0$ such that for any open set $ U$ we can find $ g \in G$ such that $ g.U$ has diameter greater than $ \varepsilon.$ We prove that if a countable $ G$ 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 $ G$-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.

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