# How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency

Gorodetski A, Kaloshin V. 2007. How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. Advances in Mathematics. 208(2), 710–797.

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*Journal Article*|

*Published*|

*English*

Author

Gorodetski, A.;
Kaloshin, Vadim

^{ISTA}^{}Abstract

Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S2, in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191–202; S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9–18; S. Newhouse, The abundance of
wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Études Sci.
50 (1979) 101–151]. It turns out that in the space of Cr smooth diffeomorphisms Diffr(M) of a compact
surface M there is an open set U such that a Baire generic diffeomorphism f ∈ U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism
have infinitely many sinks.” Our main result roughly says that with probability one for any positive D a
surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those
whose period is relatively large compared to its cyclicity. It verifies a particular case of Palis’ Conjecture
saying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have
probability zero.
One of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number
of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched
exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II,
preprint, 85 pp.].

Keywords

Publishing Year

Date Published

2007-01-30

Journal Title

Advances in Mathematics

Volume

208

Issue

2

Page

710-797

ISSN

IST-REx-ID

### Cite this

Gorodetski A, Kaloshin V. How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency.

*Advances in Mathematics*. 2007;208(2):710-797. doi:10.1016/j.aim.2006.03.012Gorodetski, A., & Kaloshin, V. (2007). How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency.

*Advances in Mathematics*. Elsevier. https://doi.org/10.1016/j.aim.2006.03.012Gorodetski, A., and Vadim Kaloshin. “How Often Surface Diffeomorphisms Have Infinitely Many Sinks and Hyperbolicity of Periodic Points near a Homoclinic Tangency.”

*Advances in Mathematics*. Elsevier, 2007. https://doi.org/10.1016/j.aim.2006.03.012.A. Gorodetski and V. Kaloshin, “How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency,”

*Advances in Mathematics*, vol. 208, no. 2. Elsevier, pp. 710–797, 2007.Gorodetski, A., and Vadim Kaloshin. “How Often Surface Diffeomorphisms Have Infinitely Many Sinks and Hyperbolicity of Periodic Points near a Homoclinic Tangency.”

*Advances in Mathematics*, vol. 208, no. 2, Elsevier, 2007, pp. 710–97, doi:10.1016/j.aim.2006.03.012.