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A $ C^2$ generic trichotomy for diffeomorphisms: Hyperbolicity or zero Lyapunov exponents or the $ C^1$ creation of homoclinic bifurcations


Author: Shuhei Hayashi
Journal: Trans. Amer. Math. Soc. 366 (2014), 5613-5651
MSC (2010): Primary 37C20, 37D20, 37D25, 37D30, 37G25
Published electronically: July 25, 2014
MathSciNet review: 3256177
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Abstract: Palis conjectured that densely in $ \mbox {Diff}^r(M)$, $ r \ge 1$, diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations. We prove a generic trichotomy for $ C^2$ diffeomorphisms: an Axiom A diffeomorphism with no cycles or Kupka-Smale ones admitting zero Lyapunov exponents or the $ C^1$ creation of homoclinic bifurcations (i.e., the creation of homoclinic tangencies or heterodimensional cycles by some $ C^1$ small perturbations).


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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/S0002-9947-2014-06425-8
Keywords: Axiom A with no cycles, heterodimensional cycles, homoclinic tangencies, Lyapunov exponents, hyperbolic measures, Pesin theory, $C^2$ generic properties
Received by editor(s): January 14, 2012
Published electronically: July 25, 2014
Article copyright: © Copyright 2014 American Mathematical Society