Topological entropy of homoclinic closures
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- by Leonardo Mendoza PDF
- Trans. Amer. Math. Soc. 311 (1989), 255-266 Request permission
Abstract:
In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in $n$-dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen’s one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.References
- Rufus Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. MR 0482842
- M. Gromov, Entropy, homology and semialgebraic geometry, Astérisque 145-146 (1987), 5, 225–240. Séminaire Bourbaki, Vol. 1985/86. MR 880035 A. B. Katok, Lyapunov enponents, entropy and periodic points, Inst. Hautes Études. Sci. Publ. Math. 51 (1980), 137-173. A. B. Katok and L. Mendoza, Smooth ergodic theory, in preparation.
- Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
- Sheldon E. Newhouse, Lectures on dynamical systems, Dynamical systems (Bressanone, 1978) Liguori, Naples, 1980, pp. 209–312. MR 660646 —, Entropy and volume, Preprint.
- Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
- Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285–300. MR 889979, DOI 10.1007/BF02766215
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 311 (1989), 255-266
- MSC: Primary 58F11; Secondary 28D20, 58F15
- DOI: https://doi.org/10.1090/S0002-9947-1989-0974777-0
- MathSciNet review: 974777