# American Institute of Mathematical Sciences

May  2012, 32(5): 1657-1674. doi: 10.3934/dcds.2012.32.1657

## Minimal skew products with hypertransitive or mixing properties

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

Received  December 2010 Revised  August 2011 Published  January 2012

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.
Citation: Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
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##### References:
 [1] S. Agronsky and J. G. Ceder, Each Peano subspace of $E^k$ is an $\omega$-limit set,, Real Anal. Exchange, 17 (): 371.   Google Scholar [2] Ll. Alsedà, S. Kolyada, J. Llibre and Ľ. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2.  Google Scholar [3] F. Balibrea and Ľ. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239. doi: 10.1016/S0166-8641(03)00090-7.  Google Scholar [4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar [5] M. Dirbák, Extensions of dynamical systems without increasing the entropy, Nonlinearity, 21 (2008), 2693-2713. doi: 10.1088/0951-7715/21/11/011.  Google Scholar [6] M. Dirbák and P. Maličký, On the construction of non-invertible minimal skew products, J. Math. Anal. Appl., 375 (2011), 436-442. doi: 10.1016/j.jmaa.2010.09.042.  Google Scholar [7] D. van Dantzig and B. L. van der Waerden, Über metrisch homogene Räume, Abhandlungen Hamburg, 6 (1928), 367-376. Google Scholar [8] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math., 34 (1979), 321-336. doi: 10.1007/BF02760611.  Google Scholar [9] K. H. Hofmann and S. A. Morris, "The Structure of Compact Groups. A Primer for the Student--a Handbook for the Expert," Second revised and augmented edition, de Gruyter Studies in Mathematics, 25, Walter de Gruyter & Co., Berlin, 2006.  Google Scholar [10] S. Kolyada and M. Matviichuk, On extensions of transitive maps, Discrete Contin. Dyn. Syst., 30 (2011), 767-777.  Google Scholar [11] S. Kolyada, Ľ. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168 (2001), 141-163. doi: 10.4064/fm168-2-5.  Google Scholar [12] K. Kuratowski, "Topology," Vol. I, New edition, revised and augmented, Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966.  Google Scholar
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