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August  2011, 30(3): 767-777. doi: 10.3934/dcds.2011.30.767

On extensions of transitive maps

Sergiĭ Kolyada 1, and  Mykola Matviichuk 2,

1. 

Institute of Mathematics, NASU, Tereshchenkivs'ka 3, 01601 Kyiv, Ukraine
2. 

National Taras Shevchenko University of Kyiv, Faculty of Mechanics and Mathematics, bul. 7, 2, Academician Glushkov pr., 03127, Kyiv, Ukraine

Received  August 2009 Revised  January 2011 Published  March 2011

For a continuous selfmap $f$ of a compact metric space $X$ we study the set of its continuous extensions $F$ on the space $X\times I$, where $I$ is a compact interval. In particular, we have solved an open problem (raised in [Ll. Alseda, S. Kolyada, J. Llibre, and L. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999)]) by proving that any continuous transitive map $f$ on $X$ can be extended to a continuous transitive triangular map $F=(f,g_x)$ on $X\times I$ without increasing topological entropy.
Keywords: triangular maps, topological entropy, Topologically transitive maps, weakly incompressible map., topologically weakly mixing maps.
Mathematics Subject Classification: Primary: 37B40; Secondary: 37B0.
Citation: Sergiĭ Kolyada, Mykola Matviichuk. On extensions of transitive maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 767-777. doi: 10.3934/dcds.2011.30.767
References:
[1]

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

[2]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms,, Trans. Moscow. Math. Soc., 23 (1970), 1. Google Scholar

[3]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems,, Proc. London Math. Soc., 3 (1954), 168. doi: 10.1112/plms/s3-4.1.168. Google Scholar

[4]

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

[5]

A. Fathi, Skew products and minimal dynamical systems on separable Hilbert manifolds,, Ergodic Theory Dynam. Systems, 4 (1984), 213. doi: 10.1017/S014338570000239X. Google Scholar

[6]

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

[7]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems,, Random Comput. Dynam., 4 (1996), 205. Google Scholar

[8]

A. N. Sharkovskiĭ, Continuous mapping on the limit points of an iteration sequence,, (Russian) Ukrain. Mat. Zh., 18 (1966), 127. Google Scholar

[9]

M. Stefankova, On topological entropy of transitive triangular maps,, Topology Appl., 153 (2006), 2673. doi: 10.1016/j.topol.2005.11.002. Google Scholar

show all references

References:
[1]

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

[2]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms,, Trans. Moscow. Math. Soc., 23 (1970), 1. Google Scholar

[3]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems,, Proc. London Math. Soc., 3 (1954), 168. doi: 10.1112/plms/s3-4.1.168. Google Scholar

[4]

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

[5]

A. Fathi, Skew products and minimal dynamical systems on separable Hilbert manifolds,, Ergodic Theory Dynam. Systems, 4 (1984), 213. doi: 10.1017/S014338570000239X. Google Scholar

[6]

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

[7]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems,, Random Comput. Dynam., 4 (1996), 205. Google Scholar

[8]

A. N. Sharkovskiĭ, Continuous mapping on the limit points of an iteration sequence,, (Russian) Ukrain. Mat. Zh., 18 (1966), 127. Google Scholar

[9]

M. Stefankova, On topological entropy of transitive triangular maps,, Topology Appl., 153 (2006), 2673. doi: 10.1016/j.topol.2005.11.002. Google Scholar

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