Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Applications of Mañé's $ C^2$ connecting lemma

Author(s): Shuhei Hayashi
Journal: Proc. Amer. Math. Soc. 138 (2010), 1371-1385.
MSC (2010): Primary 37C20, 37C29, 37C40, 37D05, 37D20
Posted: December 4, 2009
MathSciNet review: 2578529
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We consider a few applications of Mañé's $ C^2$ Connecting Lemma. These are the $ C^2$ creation of homoclinic points associated to a basic set (i.e., isolated transitive hyperbolic set), a $ C^2$ 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 $ C^2$ generically uniformly hyperbolic diffeomorphisms.

References:

[H1]
S. Hayashi, Diffeomorphisms in $ {\mathcal F}^1(M)$ satisfy Axiom A, Ergod. Th. and Dynam. Sys. 12 (1992), 233-253. MR 1176621 (94d:58081)

[H2]
S. Hayashi, Connecting invariant manifolds and the solution of the $ C^1$ stability and $ \Omega$-stability conjectures for flows, Ann. of Math. (2) 145 (1997), 81-137. MR 1432037 (98b:58096)

[M1]
R. Mañé, On the creation of homoclinic points, Publ. Math. I.H.E.S. 66 (1988), 139-159. MR 932137 (89e:58089)

[M2]
R. Mañé, A proof of the $ C^1$ stability conjecture, Publ. Math. I.H.E.S. 66 (1988), 161-210. MR 932138 (89e:58090)

[O]
F. Oliveira, On the generic existence of homoclinic points, Ergod. Th. and Dynam. Sys. 7 (1987), 567-595. MR 922366 (89j:58104)

[P]
J. Palis, A note on $ \Omega$-stability, Proc. Symp. in Pure Math., 14, Amer. Math. Soc., 1970, 221-222. MR 0270387 (42:5276)

[PdM]
J. Palis, W. de Melo, Geometric theory of dynamical systems. An Introduction, Springer-Verlag, 1982. MR 669541 (84a:58004)

[Pi]
D. Pixton, Planar homoclinic points, J. Diff. Eq. 44 (1982), no. 3, 365-382. MR 661158 (83h:58077)

[Pu]
E. Pujals, Some simple questions related to the $ C^r$ stability conjecture, Nonlinearity 21 (2008), T233-T237. MR 2448223

[R]
C. Robinson, Closing stable and unstable manifolds on the two-sphere, Proc. Amer. Math. Soc. 41 (1973), 299-303. MR 0321141 (47:9674)

[S]
M. Shub, Global stability of dynamical systems, Springer-Verlag, 1987. MR 869255 (87m:58086)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37C20, 37C29, 37C40, 37D05, 37D20

Retrieve articles in all Journals with MSC (2010): 37C20, 37C29, 37C40, 37D05, 37D20

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: 10.1090/S0002-9939-09-10148-X
PII: S 0002-9939(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
Posted: December 4, 2009
Communicated by: Bryna Kra
Copyright of article: Copyright 2009, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia