New Criteria of Generic Hyperbolicity based on Periodic Points
Armando Castro
备注：
22 pages. This new version, published in Bull. of Braz. Math. Society in 2011, just covers the preprints arXiv:0906.2240 and ArXiv 0906.2031 published by the author in ArXiv in June 12, 2009. These results were announced by the author for the first time in the 2008 Summer School of Dyn. Systems, at ICTP, Trieste, Italy
We prove a criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild (non uniform) hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a $C^1$-open set $\SU$ then there exists an open and dense subset $\SA\subset \SU$ of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms. As a simple corollary of our techniques, we have that an arbitrary $\mathrm{C}^1$-class local diffeomorphism $f$ of a closed manifold $M^n$ is uniformly expanding on the closure $\mathrm{Cl}_{M^n}(\mathrm{Per}(f))$ of its periodic point set $\mathrm{Per}(f)$, if it is nonuniformly expanding on $\mathrm{Per}(f)$.
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Peking University（北京大学）；Queen Mary, University of London（伦敦大学玛丽女王学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
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Florida State University（佛罗里达州立大学）；Monash University（蒙纳士大学）；The Technion IIT（以色列理工学院）；Facultad de Ingeniería, Universidad de la República（乌拉圭共和国大学工程学院）
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For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C1 perturbations and global structures for the dynamics ( such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a difículty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. In [BdL] with Christan Bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities. In this paper, using this tool, we recover the results in [BDP] for flows, showing that robustly cha…
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Generalized hyperbolicity for diffeomorphisms of Banach spaces
Pontificia Universidade Catolica do Rio de Janeiro (PUC-Rio)（天主教里约热内卢 Pontifical University）；Institute of Mathematics, Statistics and Computer Science, University of São Paulo (IME-USP)（圣保罗大学数学、统计和计算机科学学院）
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