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On topological entropy of group actions on $ S\sp 1$

Author: Nobuya Watanabe
Journal: Proc. Amer. Math. Soc. 106 (1989), 245-249
MSC: Primary 58F11; Secondary 57S25, 58F18
MathSciNet review: 965946
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Abstract: In this paper, we show that a surface group action on $ {S^1}$ with a non-zero Euler number has a positive topological entropy. We also show that if a surface group action on $ {S^1}$ has a Euler number which attains the maximal absolute value in the inequality of Milnor-Wood, then the topological entropy of the action equals the exponential growth rate of the group.

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  • [B] R. Bowen, Hausdorf dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1970), 11-25. MR 556580 (81g:57023)
  • [C-C] J. Cantwell and L. Conlon, Nonexponential leaves at finite level, Trans. Amer. Math. Soc. 269 (1982), 637-661. MR 637715 (84h:57013)
  • [Gh] E. Ghys, Groupes d'homeomorphismes du cercle et cohomologie bornee, The Lefschetz centennial conference. Part 111: Proceedings on differential equations, Contemporary Math., vol. 58, Amer. Math. Soc., Providence, R. I., 1987, 81-106. MR 893858 (88m:58024)
  • [G-L-W] E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math. 160 (1988), 105-142. MR 926526 (89a:57034)
  • [Go] W. M. Goldman, Discontinuous groups and the Euler class, Thesis, Berkley.
  • [Gr] M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.E.S. 56 (1982), 5-100. MR 686042 (84h:53053)
  • [He] G. Hector, Leaves whose growth is neither exponential nor polynomial, Topology 16 (1977), 451-459. MR 0464260 (57:4194)
  • [Hu] S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder, preprint, I.H.E.S.. MR 970562 (89k:58155)
  • [Ma-1] S. Matsumoto, Numerical invariants for semiconjugacy of homeomorphisms of the circle, Proc. Amer. Math. Soc. 98 (1986), 163-168. MR 848896 (88b:58121)
  • [Ma-2] -, Some remarks on foliated $ {S^1}$ bundle, Invent. Math. 90 (1987), 343-358. MR 910205 (88k:58016)
  • [Mi] J. Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215-223. MR 0095518 (20:2020)
  • [S] C. Series, Geometrical Markov coding of geodesies on surfaces of constant negative curvature, Ergod. Th. and Dynam. Sys. 6 (1986), 601-625. MR 873435 (88k:58130)
  • [Wa] P. Walters, An introduction to ergodic theory, GTM, 79, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)
  • [Wo] J. Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971), 257-273. MR 0293655 (45:2732)

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Keywords: Topological entropy, surface group action, Euler number
Article copyright: © Copyright 1989 American Mathematical Society

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