Sensitive dependence on initial conditions and chaotic group actions
Author:
- Fabrizio PoloFPFabrizio Polo
Preprint
Abstract
A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number e>0 such that for any open set U we can find g in G such that g.U has diameter greater than e. We prove that if a G action preserves a probability measure of full support, then the system is either minimal and equicontinuous, or has sensitive dependence on initial conditions. This generalizes the invertible case of a theorem of Glasner and Weiss. We prove that when a finitely generated, solvable group, acts and certain cyclic subactions have dense sets of minimal points, then 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, group actions.Keywords
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This publication has 5 references indexed in Scilit:
- A note on sensitivity of semigroup actionsSemigroup Forum, 2007
- Residual properties and almost equicontinuityJournal d'Analyse Mathématique, 2001
- Interval maps, factors of maps, and chaosTohoku Mathematical Journal, 1980
- Sensitive dependence to initial conditions for one dimensional mapsCommunications in Mathematical Physics, 1979
- Some remarks on monothetic groupsColloquium Mathematicum, 1964