Hyperbolicity of localizations

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.

Localizations of Groups

A group homomorphism η : A → H is called a localization of A if every homomorphism φ : A → H can be ‘extended uniquely’ to a homomorphism Φ : H → H in the sense that Φη = φ. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the intr...

Beyond Hyperbolicity

Hyperbolicity of Unitary Involutions

We prove the so-called Unitary Hyperbolicity Theorem, a result on hyperbolicity of unitary involutions. The analogous previously known results for the orthogonal and symplectic involutions are formal consequences of the unitary one. While the original proofs in the orthogonal and symplectic cases were based on the incompressibility of generalized Severi-Brauer varieties, the proof in the unitar...