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 - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
References:
[1]

S. Agronsky and J. G. Ceder, Each Peano subspace of $E^k$ is an $\omega$-limit set,, Real Anal. Exchange, 17 (): 371.

[2]

Ll. Alsedà, S. Kolyada, J. Llibre and Ľ. Snoha, Entropy and periodic points for transitive maps,, Trans. Amer. Math. Soc., 351 (1999), 1551. doi: 10.1090/S0002-9947-99-02077-2.

[3]

F. Balibrea and Ľ. Snoha, Topological entropy of Devaney chaotic maps,, Topology Appl., 133 (2003), 225. doi: 10.1016/S0166-8641(03)00090-7.

[4]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[5]

M. Dirbák, Extensions of dynamical systems without increasing the entropy,, Nonlinearity, 21 (2008), 2693. doi: 10.1088/0951-7715/21/11/011.

[6]

M. Dirbák and P. Maličký, On the construction of non-invertible minimal skew products,, J. Math. Anal. Appl., 375 (2011), 436. doi: 10.1016/j.jmaa.2010.09.042.

[7]

D. van Dantzig and B. L. van der Waerden, Über metrisch homogene Räume,, Abhandlungen Hamburg, 6 (1928), 367.

[8]

S. Glasner and B. Weiss, On the construction of minimal skew products,, Israel J. Math., 34 (1979), 321. doi: 10.1007/BF02760611.

[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 (2006).

[10]

S. Kolyada and M. Matviichuk, On extensions of transitive maps,, Discrete Contin. Dyn. Syst., 30 (2011), 767.

[11]

S. Kolyada, Ľ. Snoha and S. Trofimchuk, Noninvertible minimal maps,, Fund. Math., 168 (2001), 141. doi: 10.4064/fm168-2-5.

[12]

K. Kuratowski, "Topology," Vol. I, New edition, revised and augmented,, Translated from the French by J. Jaworowski, (1966).

show all references

References:
[1]

S. Agronsky and J. G. Ceder, Each Peano subspace of $E^k$ is an $\omega$-limit set,, Real Anal. Exchange, 17 (): 371.

[2]

Ll. Alsedà, S. Kolyada, J. Llibre and Ľ. Snoha, Entropy and periodic points for transitive maps,, Trans. Amer. Math. Soc., 351 (1999), 1551. doi: 10.1090/S0002-9947-99-02077-2.

[3]

F. Balibrea and Ľ. Snoha, Topological entropy of Devaney chaotic maps,, Topology Appl., 133 (2003), 225. doi: 10.1016/S0166-8641(03)00090-7.

[4]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[5]

M. Dirbák, Extensions of dynamical systems without increasing the entropy,, Nonlinearity, 21 (2008), 2693. doi: 10.1088/0951-7715/21/11/011.

[6]

M. Dirbák and P. Maličký, On the construction of non-invertible minimal skew products,, J. Math. Anal. Appl., 375 (2011), 436. doi: 10.1016/j.jmaa.2010.09.042.

[7]

D. van Dantzig and B. L. van der Waerden, Über metrisch homogene Räume,, Abhandlungen Hamburg, 6 (1928), 367.

[8]

S. Glasner and B. Weiss, On the construction of minimal skew products,, Israel J. Math., 34 (1979), 321. doi: 10.1007/BF02760611.

[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 (2006).

[10]

S. Kolyada and M. Matviichuk, On extensions of transitive maps,, Discrete Contin. Dyn. Syst., 30 (2011), 767.

[11]

S. Kolyada, Ľ. Snoha and S. Trofimchuk, Noninvertible minimal maps,, Fund. Math., 168 (2001), 141. doi: 10.4064/fm168-2-5.

[12]

K. Kuratowski, "Topology," Vol. I, New edition, revised and augmented,, Translated from the French by J. Jaworowski, (1966).

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