Nondegenerate homoclinic tangency and hyperbolic sets
type: Item
instance of: scholarly article
title: Nondegenerate homoclinic tangency and hyperbolic sets
zbMATH Open document ID: 1023.37012
DOI: 10.1016/S0362-546X(02)00271-7
author: Ming-Chia Li
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The paper shows that a nondegenerate homoclinic tangency on a surface is accumulated by a sequence of uniformly hyperbolic invariant sets and it is a boundary point of a nonuniformly hyperbolic invariant set. Main results of the paper are the following two theorems: Theorem 1. Let \(q\) be a nondegenerate homoclinic tangency for a dissipative periodic saddle point \(p\) of a diffeomorphism \(f\) on a two-dimensional manifold. Then there is a positive integer \(N\) such that, for any integer \(n\geq N\), there exists a region \(B_n\) such that the invariant set \(\Lambda _n=\bigcap _{j=-\infty} ^{\infty}f^{jn}(B_n)\) has a uniformly hyperbolic structure for \(f^n\). Moreover, \(q\) lies in the closure of \(\bigcup _{n\geq N}\Lambda _n\). Theorem 2. Under the same assumptions as in Theorem 1, there exists a region \(B\) such that if \(F\) is the first return map of \(f\) on \(B\) and \(\Lambda _B=\bigcap _{j=-\infty} ^{\infty}F^{j}B\) then \(\widetilde\Lambda =\bigcup _{i\geq 0}f^{i}(\Lambda _B)\) has a nonuniformly hyperbolic structure for \(f\). Furthermore, both \(p\) and \(q\) are on the boundary of \(\widetilde\Lambda\). The obtained results are related to the Hénon map \(H(x,y)=(a-by-x^2,x)\) with \(|b|\) relatively large and to the Smale horseshoe with homoclinic tangency.
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37E30
zbMATH DE Number: 1880109
zbMATH Keywords: surface diffeomorphism
zbMATH Keywords: nondegenerate homoclinic tangency
zbMATH Keywords: uniform and nonuniformly hyperbolic sets
zbMATH Keywords: dissipative periodic saddle point
zbMATH Keywords: cone field
zbMATH Keywords: Hénon map
reviewed by: Eugene Ershov
MaRDI profile type: Publication
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full work available at URL: s0362-546x(02)00271-7
OpenAlex ID: W2026104157
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description: scientific article; zbMATH DE number 1880109