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Hyperbolicity and the creation of homoclinic orbits. (English) Zbl 0641.58029
Authors’ summary: “We consider one-parameter families \(\{\phi_{\mu}: \mu \in {\mathbb{R}}\}\) of diffeomorphisms on surfaces which display a homoclinic tangency for \(\mu =0\) and are hyperbolic for \(\mu <0\) (i.e. \(\phi_{\mu}\) has a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for \(\mu\) positive. For many of these families, we prove that \(\phi_{\mu}\) is also hyperbolic for most small positive values of \(\mu\) (which implies much regularity of the dynamical structure). A main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency.”
The main result extends some previous results of S. Newhouse and the first author [Astérisque 31, 44-140 (1976; Zbl 0322.58009)] and the authors [Invent. Math. 82, 397-422 (1985; Zbl 0579.58005)], and its proof uses some arguments from the latter paper.
Reviewer: L.N.Stoyanov

MSC:
37D99 Dynamical systems with hyperbolic behavior
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