TAILIEUCHUNG - Đề tài " A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources "

A C 1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources By C. Bonatti, L. J. D´ ıaz, and E. R. Pujals* A Ricardo Ma˜e (1948–1995), por todo su trabajo n´ Abstract We show that, for every compact n-dimensional manifold, n ≥ 1, there is a residual subset of Diff1 (M ) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of. | Annals of Mathematics A C1-generic dichotomy for diffeomorphisms Weak forms of hyperbolicity or infinitely many sinks or sources By C. Bonatti L. J. Diaz and E. R. Pujals Annals of Mathematics 158 2003 355 418 A C generic dichotomy for diffeomorphisms Weak forms of hyperbolicity or infinitely many sinks or sources By C. Bonatti L. J. Diaz and E. R. Pujals A Ricardo Mane 1948-1995 por todo su trabajo Abstract We show that for every compact n-dimensional manifold n 1 there is a residual subset of Diff1 M of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities Either it is contained in the closure of an infinite set of sinks or sources Newhouse phenomenon or it presents some weak form of hyperbolicity called dominated splitting this is a generalization of a bidimensional result of Mane Ma3 . In particular we show that any C 1-robustly transitive diffeomorphism admits a dominated splitting. Resume Generalisant un resultat de Mane sur les surfaces Ma3 nous montrons que en dimension quelconque il existe un sous-ensemble residuel de Diff1 M de diffeomorphismes pour lesquels la classe homocline de toute selle periodique hyperbolique possede deux comportements possibles ou bien elle est incluse dans l adherence d une infinite de puits ou de sources phenomene de Newhouse ou bien elle presente une forme affaiblie d hyperbolicite appelee une decomposition dominée. En particulier nous montrons que tout diffeomorphisme C 1-robustement transitif possede une decomposition dominee. Introduction Context. The Anosov-Smale theory of uniformly hyperbolic systems has played a double role in the development of the qualitative theory of dynamical systems. On one hand this theory shows that chaotic and random behavior can appear in a stable way for deterministic systems depending on a very small number of parameters. On the other hand the chaotic systems admit in this Partially supported by CNPQ FAPERJ IMPA and .

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