×

zbMATH — the first resource for mathematics

Generic robustness of spectral decompositions. (English. French summary) Zbl 1027.37010
Let \(f\in \text{Diff}^1(M)\), where \(M\) is a compact boundaryless manifold of dimension \(n\geq 2\) and \(p\in M\) is a periodic hyperbolic saddle of \(f\). The homoclinic class of \(f\) relative to \(p\) is given by \(H(p,f)= \text{cl}[W^s(p)\pitchfork W^u(p)]\), where \(\pitchfork\) denotes points of transverse intersection of the invariant manifolds. Homoclinic classes are the natural candidates to replace hyperbolic basic sets in nonhyperbolic theory.
The main result of the paper says that there exists a residual subset \(R\) of \(\text{Diff}^1(M)\) such that if \(f\in R\) has only finitely many distinct homoclinic classes \(H(p_1,f),\dots, H(p_k,f)\), then there exists a neighborhood \(U\) of \(f\) in \(R\) such that if \(g\in U\), then the only distinct homoclinic classes of \(g\) are the continuations \(H(p^g_1, g),\dots, H(p^g_k, g)\) of the homoclinic classes of \(f\). Further there exists a residual subset \(\widetilde R\) of \(\text{Diff}^1(M)\) such that if \(f\in\widetilde R\) admits a spectral decomposition, then this one is generically robust. The aforementioned results are used to prove the Palis conjecture, which states that there exists a dense subset \(D\) such that if \(f\in D\), then \(f\) either is hyperbolic or exhibits homoclinic bifurcations.

MSC:
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37G30 Infinite nonwandering sets arising in bifurcations of dynamical systems
37C29 Homoclinic and heteroclinic orbits for dynamical systems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Abdenur F., Attractors of generic diffeomorphisms are persistent , preprint IMPA, 2001. MR 1950789
[2] Bonatti Ch. , Diaz L.J. , Persistence of transitive diffeomorphisms , Ann. Math. 143 ( 1995 ) 367 - 396 . MR 1381990 | Zbl 0852.58066 · Zbl 0852.58066 · doi:10.2307/2118647
[3] Bonatti Ch. , Diaz L.J. , Connexions hétéroclines et généricité d’une infinité de puits ou de sources , Ann. Scient. Éc. Norm. Sup. Paris 32 ( 1999 ) 135 - 150 . Numdam | MR 1670524 | Zbl 0944.37012 · Zbl 0944.37012 · doi:10.1016/S0012-9593(99)80012-3 · numdam:ASENS_1999_4_32_1_135_0
[4] Bonatti Ch ., Diaz L.J., On maximal transitive sets of generic diffeomorphisms , preprint PUC-Rio, 2001.
[5] Bonatti Ch. , Diaz L.J., Pujals E., A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. Math. , to appear. MR 2018925 | Zbl 1049.37011 · Zbl 1049.37011 · doi:10.4007/annals.2003.158.355 · euclid:annm/1069786250
[6] Bonatti Ch. , Diaz L.J., Pujals E., Rocha J., Robustly transitive sets and heterodimensional cycles, Astérisque , to appear. MR 2052302 | Zbl 1056.37024 · Zbl 1056.37024
[7] Bonatti Ch. , Viana M. , SRB measures for partially hyperbolic systems whose central direction is mostly contracting , Israel J. Math. 115 ( 2000 ) 157 - 193 . MR 1749677 | Zbl 0996.37033 · Zbl 0996.37033 · doi:10.1007/BF02810585
[8] Carballo C.M., Morales C.A., Homoclinic classes and finitude of attractors for vector fields on n-manifolds , preprint, 2001. arXiv | MR 1934436 · Zbl 1035.37007 · minidml.mathdoc.fr
[9] Carballo C.M., Morales C.A., Pacifico M.J., Homoclinic classes for generic C 1 vector fields, Ergodic Theory Dynam. Systems , to appear. MR 1972228 | Zbl 1047.37009 · Zbl 1047.37009 · doi:10.1017/S0143385702001116
[10] Diaz L.J. , Pujals E. , Ures R. , Partial hyperbolicity and robust transitivity , Acta Math. 183 ( 1999 ) 1 - 43 . MR 1719547 | Zbl 0987.37020 · Zbl 0987.37020 · doi:10.1007/BF02392945
[11] Franks J. , Necessary conditions for stability of diffeomorphisms , Trans. AMS 158 ( 1971 ) 301 - 308 . MR 283812 | Zbl 0219.58005 · Zbl 0219.58005 · doi:10.2307/1995906
[12] Hayashi S. , Diffeomorphisms in I 1 (M) satisfy Axiom A , Ergodic Theory Dynam. Systems 12 ( 1992 ) 233 - 253 . MR 1176621 | Zbl 0760.58035 · Zbl 0760.58035 · doi:10.1017/S0143385700006726
[13] Hayashi S. , Connecting invariant manifolds and the solution of the C 1 stability and \Omega -stability conjectures for flows , Ann. Math. 145 ( 1997 ) 81 - 137 . Zbl 0871.58067 · Zbl 0871.58067 · doi:10.2307/2951824 · eudml:129684
[14] Kelley J.L. , General Topology , New York , Springer , 1955 . MR 70144 | Zbl 0306.54002 · Zbl 0306.54002
[15] Mañé R. , Contributions to the C 1 -stability conjecture , Topology 17 ( 1978 ) 386 - 396 . MR 516217 | Zbl 0405.58035 · Zbl 0405.58035 · doi:10.1016/0040-9383(78)90005-8
[16] Mañé R. , An ergodic closing lemma , Ann. Math. 116 ( 1982 ) 503 - 540 . MR 678479 | Zbl 0511.58029 · Zbl 0511.58029 · doi:10.2307/2007021
[17] Palis J. , A global view of dynamics and a conjecture on the denseness of finitude of atttractors , Astérisque 261 ( 2000 ) 335 - 347 . MR 1755446 | Zbl 1044.37014 · Zbl 1044.37014
[18] Pugh C. , An improved closing lemma and a general density theorem , Amer. J. Math. 89 ( 1967 ) 1010 - 1021 . MR 226670 | Zbl 0167.21804 · Zbl 0167.21804 · doi:10.2307/2373414
[19] Pujals E. , Sambarino M. , Homoclinic tangencies and hyperbolicity for surface diffeomorphisms: a conjecture of Palis , Ann. Math. 151 ( 2000 ) 961 - 1023 . MR 1779562 | Zbl 0959.37040 · Zbl 0959.37040 · doi:10.2307/121127 · eudml:120908
[20] Palis J. , Takens F. , Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations , Cambridge Univ. Press , 1993 . MR 1237641 | Zbl 0790.58014 · Zbl 0790.58014
[21] Shub M. , Global Stability of Dynamical Systems , Springer-Verlag , New York , 1986 . MR 869255 | Zbl 0606.58003 · Zbl 0606.58003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.