On extensions of transitive maps

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2.
[2]	D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow. Math. Soc., 23 (1970), 1-35.
[3]	Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., 3 (1954), 168-176. doi: 10.1112/plms/s3-4.1.168.
[4]	M. Dirbak, Extensions of dynamical systems without increasing the entropy, Nonlinearity, 21 (2008), 2693-2713. doi: 10.1088/0951-7715/21/11/011.
[5]	A. Fathi, Skew products and minimal dynamical systems on separable Hilbert manifolds, Ergodic Theory Dynam. Systems, 4 (1984), 213-224. doi: 10.1017/S014338570000239X.
[6]	S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math., 34 (1979), 321-336. doi: 10.1007/BF02760611.
[7]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233.
[8]	A. N. Sharkovskiĭ, Continuous mapping on the limit points of an iteration sequence, (Russian) Ukrain. Mat. Zh., 18 (1966), 127-130.
[9]	M. Stefankova, On topological entropy of transitive triangular maps, Topology Appl., 153 (2006), 2673-2679. doi: 10.1016/j.topol.2005.11.002.

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), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2.
[2]	D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow. Math. Soc., 23 (1970), 1-35.
[3]	Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., 3 (1954), 168-176. doi: 10.1112/plms/s3-4.1.168.
[4]	M. Dirbak, Extensions of dynamical systems without increasing the entropy, Nonlinearity, 21 (2008), 2693-2713. doi: 10.1088/0951-7715/21/11/011.
[5]	A. Fathi, Skew products and minimal dynamical systems on separable Hilbert manifolds, Ergodic Theory Dynam. Systems, 4 (1984), 213-224. doi: 10.1017/S014338570000239X.
[6]	S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math., 34 (1979), 321-336. doi: 10.1007/BF02760611.
[7]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233.
[8]	A. N. Sharkovskiĭ, Continuous mapping on the limit points of an iteration sequence, (Russian) Ukrain. Mat. Zh., 18 (1966), 127-130.
[9]	M. Stefankova, On topological entropy of transitive triangular maps, Topology Appl., 153 (2006), 2673-2679. doi: 10.1016/j.topol.2005.11.002.

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