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

## On extensions of transitive maps

 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.
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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