Minimal skew products with hypertransitive or mixing properties

Matúš Dirbák

doi:10.3934/dcds.2012.32.1657

Article Contents

2012, Volume 32, Issue 5: 1657-1674. Doi: 10.3934/dcds.2012.32.1657

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Minimal skew products with hypertransitive or mixing properties

Matúš Dirbák^1,

1.
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica

Received: December 2010

Revised: August 2011

Published: May 2012

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Abstract

Let $X$ be an infinite compact metric space and let $Z$ be a compact metric space admitting an arc-wise connected group $\mathcal H_0(Z)$ of homeomorphisms whose natural action on $Z$ is topologically transitive. We show that every map $f$ on $X$ with a hypertransitive property $\Lambda$ admits a skew product extension $F=(f,g_x)$ on $X\times Z$ which also has the property $\Lambda$ and whose all fibre maps $g_x$ lie in the closure $\overline{\mathcal H_0(Z)}$ of $\mathcal H_0(Z)$ in the space $\mathcal H(Z)$ of all homeomorphisms on $Z$.
If we additionally assume that both the map $f$ and the action of $\mathcal H_0(Z)$ on $Z$ are minimal then we can guarantee the existence of such an extension $F$ in the class of minimal maps. In particular case when $\Lambda$= topological transitivity, such a theorem was known before (for invertible $f$ it was proved by Glasner and Weiss already in 1979).
Finally, we show that if one imposes further restrictions on the group $\mathcal H_0(Z)$ then the analogues of the mentioned results for hypertransitive properties $\Lambda$ hold also for $\Lambda$= strong mixing.

Keywords:

Skew product extension,
hypertransitive property,
minimality,
strong mixing,
topologically transitive and minimal group actions,
arc-wise connected group of homeomorphisms.

Mathematics Subject Classification: Primary: 37B05, 37B40; Secondary: 54H20.

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References

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