Sensitive dependence on initial conditions and chaotic group actions
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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.References
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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
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2644895