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http://dx.doi.org/10.4134/JKMS.2006.43.2.373

TOPOLOGICAL ENTROPY OF A SEQUENCE OF MONOTONE MAPS ON CIRCLES  

Zhu Yuhun (College of Mathematics and Information Science Hebei Normal University)
Zhang Jinlian (College of Mathematics and Information Science Hebei Normal University)
He Lianfa (College of Mathematics and Information Science Hebei Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.43, no.2, 2006 , pp. 373-382 More about this Journal
Abstract
In this paper, we prove that the topological entropy of a sequence of equi-continuous monotone maps $f_{1,\infty}={f_i}\;\infty\limits_{i=1}$on circles is $h(f_{1,\infty})={\frac{lim\;sup}{n{\rightarrow}\infty}}\;\frac 1 n \;log\;{\prod}\limits_{i=1}^n|deg\;f_i|$. As applications, we give the estimation of the entropies for some skew products on annular and torus. We also show that a diffeomorphism f on a smooth 2-dimensional closed manifold and its extension on the unit tangent bundle have the same entropy.
Keywords
sequence of continuous maps; topological entropy; separated set; spanning set;
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1 R. Bowen, Entropy for group endomorphisms and homogenuous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414   DOI   ScienceOn
2 E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems (Russian), Izv. Akad. Nauk. SSSR. Ser. Mat. 35 (1971), 324-366
3 D. Fiebig, U. Fiebig, and Z. Nitecki, Entropy and preimage sets, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1785-1806   DOI   ScienceOn
4 F. -H. Lian and Z. -H. Wang, Topological entropy of monotone map on circle, J. Math. Resear. Expo. 16 (1996), no. 3, 379-382
5 S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205-223
6 S. -T. Liao, Qualitative theory of differentiable dynamical systems, The Science Press of China, Beijing, 1992
7 M. Misiurewicz, On Bowen's definition of topological entropy, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 827-833   DOI
8 M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studa Math. 67 (1980), no. 1, 45-63   DOI
9 Z. Nitecki and F. Przytycki, Preimage entropy for mappings, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1815-1843   DOI
10 P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, Hei- delberg, Berlin, 1982
11 R. Adler, A. Konheim, and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319   DOI
12 S. Kolyada, M. Misiurewicz, and L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on interval. Fund. Math. 160 (1999), 161-181
13 Z. -S. Zhang, The principle of differentiable dynamical systems, The Science Press of China, Beijing, 1987