A 𝐶² generic trichotomy for diffeomorphisms: Hyperbolicity or zero Lyapunov exponents or the 𝐶¹ creation of homoclinic bifurcations

Hayashi, Shuhei

doi:10.1090/S0002-9947-2014-06425-8

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