Nondegenerate homoclinic tangency and hyperbolic sets
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.
uniform and nonuniformly hyperbolic sets
scientific article; zbMATH DE number 1880109