Hyperbolicity and the creation of homoclinic orbits


We consider one-parameter families $\{\varphi_\mu; \mu \in \mathbf{R}\}$ of diffeomorphisms on surfaces which display a homoclinic tangency for $\mu = 0$ and are hyperbolic for $\mu < 0$ (i.e., $\varphi_\mu$ has a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for $\mu$ positive. For many of these families, we prove that $\varphi_\mu$ 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.


Jacob Palis

Floris Takens