Generalized hyperbolicity

Semi-hyperbolicity and Hyperbolicity

We prove that for C1-diffeomorfisms semi-hyperbolicity of an invariant set implies its hyperbolicity. Moreover, we provide some exact estimations of hyperbolicity constants by semi-hyperbolicity ones, which can be useful in strict numerical computations.

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is δ-hyperbolic if every geodesic triangle is δ-thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize being chordal and the hyperbolicity of the graph. We also give a cha...

Pointwise Hyperbolicity Implies Uniform Hyperbolicity

We provide a general mechanism for obtaining uniform information from pointwise data. A sample result is that if a diffeomorphism of a compact Riemannian manifold has pointwise expanding and contracting continuous invariant cone families, then the diffeomorphism is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.

Beyond Hyperbolicity

Strong Hyperbolicity

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are strongly hyperbolic. On the way, we determine the best ...