Searching for 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.

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.

A Criterion for Hyperbolicity

The usual deenition of hyperbolicity of a group G demands that all geodesic triangles in the Cayley graph of G should be thin. Using the theorem that a subquadratic isoperimetric inequality implies a linear one, we show that it is in fact only necessary for all triangles from a given combing to be thin, thus giving a new criterion for hyperbolicity of nitely presented groups. Given a group G th...

Hyperbolicity Criteria for Certain Involutions

Using the ideas and techniques developed by Bayer-Fluckiger, Shapiro and Tignol about hyperbolic involutions of central simple algebras, criteria for the hyperbolicity of involutions of the form σ⊗ τ and σ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A and ρ is an involution of a quaternion algebra are obt...

Beyond Hyperbolicity