Preprint C184/2015
Sufficient Conditions for Hyperbolicity on Skew-Products
Javier Correa
Keywords: Skew-Products | Hyperbolicity | Stability

The goal of this work is to study stable sets of skew-products with dimension 1 on the fibers. By studying the continuation of the periodic points, we prove that assuming absolute stable and infinitesimal stable in the one-parameter family of perturbations associated to the uniform translation is sufficient to imply hyperbolicity. Working with bounded solution we improve the previous result assuming  H$\ddot{o}$lder variation. This means that  a set is $\alpha$-absolute stable by the uniform translation if the distance from the conjugation to the inclusion varies H$\ddot{o}$lder-continuous according to the distance of the original systems with its perturbation.  We prove that if $\alpha>1/2$, the skew-product is $C^2$ and preserves orientation on the fibers then the central direction is hyperbolic. After this we study the central topologically hyperbolic sets of Skew-Products. We see that Kupka-Smale condition and topological hyperbolicity property are not enough like it is for diffeomorphisms on surfaces (under the hypothesis of dominated splitting) or endomorphisms in dimension 1 (under the hypothesis of non critical points). Next we find an interesting family of skew-products that we will call the rigid case which has a natural way of perturbing it to obtain hyperbolicity. We finish this thesis by working on the continuation of hyperbolic periodic points proving a dichotomy for hyperbolic sets about the ambient manifold dimension.