Overlap of hyperbolicity of different types

Speaker: Katrin Gelfert 
Affiliation: UF Rio de Janeiro, Brazil

Abstract

Hyperbolic dynamical systems, by definition, present two complementary directions, one with uniform (stable) contraction and the other with uniform expansion (unstable). A famous and important example of these systems is the so-called Smale horseshoe. However, there are many systems where hyperbolicity is not satisfied. We are interested in dynamics where the effects of contractions and expansions overlap generating regions with neutral behavior. We present dynamical systems with a simple description as a skew-product of a horseshoe and two circle diffeomorphisms. Important examples are provided by the projective actions of SL(2,R) matrices. The most interesting case occurs when one matrix is hyperbolic (eigenvalues different from one) and the other is elliptic (eigenvalues of modulo one). In this model two related systems come together: a non-hyperbolic dynamical system (with stable, unstable, and central directions) and a product of matrices (called cocycle). The Lyapunov exponent (expansion rate) of the central direction of the dynamic system corresponds to the exponential growth of the norms of the products of these matrices. We want to describe the “spectrum” of such "exponents”. To do this, it is necessary to understand the underlying “thermodynamic formalism” and the structure of space of “ergodic measures”, where the occurrence of so-called non-hyperbolic measures is a key difficulty. We aim to discuss this scenario, presenting key concepts and ideas and some results.