Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological entropy of homoclinic closures

Author: Leonardo Mendoza
Journal: Trans. Amer. Math. Soc. 311 (1989), 255-266
MSC: Primary 58F11; Secondary 28D20, 58F15
MathSciNet review: 974777
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [B] R. Bowen, On Axiom A diffeomorphisms, CBMS Regional Conf. Ser. Math., no. 35, Amer. Math. Soc., Providence, R.I., 1978. MR 0482842 (58:2888)
  • [G] M. Gromov, Entropy, homology and semialgebraic geometry, Astérisque 145-6 (1987), 225-240. MR 880035 (89f:58082)
  • [K] A. B. Katok, Lyapunov enponents, entropy and periodic points, Inst. Hautes Études. Sci. Publ. Math. 51 (1980), 137-173.
  • [KM] A. B. Katok and L. Mendoza, Smooth ergodic theory, in preparation.
  • [M] J. Moser, Stable and random motions in dynamical systems, Ann. of Math. Stud., Princeton Univ. Press, Princeton, N.J., 1973. MR 0442980 (56:1355)
  • [N1] S. Newhouse, Lectures on dynamical systems, Birkhäuser, Boston, Mass., 1980. MR 660646 (83j:58042)
  • [N2] -, Entropy and volume, Preprint.
  • [P] Ya. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114. MR 0466791 (57:6667)
  • [Y] Y. Yomdin, Volume growth and entropy, Preprint, Inst. Hautes Études Sci., 1986. MR 889979 (90g:58008)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F11, 28D20, 58F15

Retrieve articles in all journals with MSC: 58F11, 28D20, 58F15

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society