In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct we provide simplified definitions of relative hyperbolicity in terms of the geometry of a Cayley graph. In particular we obtain a definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic triangle of a central left coset of peripheral subgroup.

In this paper, we address the question of the hyperbolicity and the local wellposedness of the two-layer shallow water model, with free surface, in two dimensions. We first provide a general criterion that proves the symmetrizability of this model, which implies hyperbolicity and local well-posedness in Hs(R2), with s > 2. Then, we analyze rigorously the eigenstructure associated to this model ...

The paper is a non-technical survey and is aimed to illustrate Dolgopyat’s profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat’s work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on interaction between the two – the class of dynamical systems with mixed hyperbolicity. On the one hand, this includes uniformly partially hyperbolic diffeomorphis...

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known practical algorithms for computing the hyperbolicity number of a n-vertex graph have running time O(n). Exploiting the framework of parameterized comp...

We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.

In this lecture, we will introduce the concept of hyperbolic polynomials, a generalization of real stable polynomials. We will also introduce the concept of hyperbolicity cones, which are the set of directions along which a polynomial is always real-rooted. We will prove that hyperbolicity cones are convex, study some of their properties, and invyestigate the connection between barrier argument...

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.

minimize cx such that Ax = b and x ∈ Λ+, where c ∈ R, Ax = b is a system of linear equations and Λ+ is the closure of a so called hyperbolicity cone. Hyperbolic programming generalizes semidefinite programming, but it is not known to what extent since it is not known how general the hyperbolicity cones are. The rich algebraic structure of hyperbolicity cones makes hyperbolic programming an inte...

— In this paper, we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrees of freedom Hamiltonian systems: the transversality-torsion phenomenon. keywords: Hyperbolicity, Partially hyperbolic tori, Hamiltonian systems.

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δneighborhood of the union of the other two sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) ...