Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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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)
  • Published : 2006.03.01
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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.

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References

  1. R. Adler, A. Konheim, and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319 https://doi.org/10.2307/1994177
  2. R. Bowen, Entropy for group endomorphisms and homogenuous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414 https://doi.org/10.2307/1995565
  3. E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems (Russian), Izv. Akad. Nauk. SSSR. Ser. Mat. 35 (1971), 324-366
  4. D. Fiebig, U. Fiebig, and Z. Nitecki, Entropy and preimage sets, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1785-1806 https://doi.org/10.1017/S0143385703000221
  5. F. -H. Lian and Z. -H. Wang, Topological entropy of monotone map on circle, J. Math. Resear. Expo. 16 (1996), no. 3, 379-382
  6. S. Kolyada, M. Misiurewicz, and L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on interval. Fund. Math. 160 (1999), 161-181
  7. S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205-223
  8. S. -T. Liao, Qualitative theory of differentiable dynamical systems, The Science Press of China, Beijing, 1992
  9. M. Misiurewicz, On Bowen's definition of topological entropy, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 827-833 https://doi.org/10.3934/dcds.2004.10.827
  10. M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studa Math. 67 (1980), no. 1, 45-63 https://doi.org/10.4064/sm-67-1-45-63
  11. Z. Nitecki and F. Przytycki, Preimage entropy for mappings, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1815-1843 https://doi.org/10.1142/S0218127499001309
  12. P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, Hei- delberg, Berlin, 1982
  13. Z. -S. Zhang, The principle of differentiable dynamical systems, The Science Press of China, Beijing, 1987

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