Applications of Mañé's connecting lemma

Author:
Shuhei Hayashi

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1371-1385

MSC (2010):
Primary 37C20, 37C29, 37C40, 37D05, 37D20

Published electronically:
December 4, 2009

MathSciNet review:
2578529

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a few applications of Mañé's Connecting Lemma. These are the creation of homoclinic points associated to a basic set (i.e., isolated transitive hyperbolic set), a locally generic criterion to know whether a given point belongs to the stable set of hyperbolic homoclinic classes, and that measurably hyperbolic diffeomorphisms (i.e., having the closure of supports of all invariant measures as a countable union of disjoint basic sets) are generically uniformly hyperbolic diffeomorphisms.

**[H1]**Shuhei Hayashi,*Diffeomorphisms in ℱ¹(ℳ) satisfy Axiom A*, Ergodic Theory Dynam. Systems**12**(1992), no. 2, 233–253. MR**1176621**, 10.1017/S0143385700006726**[H2]**Shuhei Hayashi,*Connecting invariant manifolds and the solution of the 𝐶¹ stability and Ω-stability conjectures for flows*, Ann. of Math. (2)**145**(1997), no. 1, 81–137. MR**1432037**, 10.2307/2951824**[M1]**Ricardo Mañé,*On the creation of homoclinic points*, Inst. Hautes Études Sci. Publ. Math.**66**(1988), 139–159. MR**932137****[M2]**Ricardo Mañé,*A proof of the 𝐶¹ stability conjecture*, Inst. Hautes Études Sci. Publ. Math.**66**(1988), 161–210. MR**932138****[O]**Fernando Oliveira,*On the generic existence of homoclinic points*, Ergodic Theory Dynam. Systems**7**(1987), no. 4, 567–595. MR**922366**, 10.1017/S0143385700004211**[P]**J. Palis,*A note on Ω-stability*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 221–222. MR**0270387****[PdM]**Jacob Palis Jr. and Welington de Melo,*Geometric theory of dynamical systems*, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR**669541****[Pi]**Dennis Pixton,*Planar homoclinic points*, J. Differential Equations**44**(1982), no. 3, 365–382. MR**661158**, 10.1016/0022-0396(82)90002-X**[Pu]**Enrique R. Pujals,*Some simple questions related to the 𝐶^{𝑟} stability conjecture*, Nonlinearity**21**(2008), no. 11, T233–T237. MR**2448223**, 10.1088/0951-7715/21/11/T02**[R]**Clark Robinson,*Closing stable and unstable manifolds on the two sphere*, Proc. Amer. Math. Soc.**41**(1973), 299–303. MR**0321141**, 10.1090/S0002-9939-1973-0321141-7**[S]**Michael Shub,*Global stability of dynamical systems*, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR**869255**

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Additional Information

**Shuhei Hayashi**

Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, Japan

Email:
shuhei@ms.u-tokyo.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-09-10148-X

Keywords:
$C^2$ generic properties,
connecting lemma,
invariant measures,
basic sets,
homoclinic points,
homoclinic classes,
uniform hyperbolicity,
Axiom A with no cycles.

Received by editor(s):
March 15, 2009

Received by editor(s) in revised form:
July 31, 2009

Published electronically:
December 4, 2009

Communicated by:
Bryna Kra

Article copyright:
© Copyright 2009
American Mathematical Society