# zbMATH — the first resource for mathematics

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
Full Text: