Jairo Bochi

Genericity of zero Lyapunov exponents

We show that, for any compact surface, there is a residual (dense Gδ )s et of C 1 area-preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponentsa.e. This result was announcedby R. Mane, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous SL(2, R)-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

A formula with some applications to the theory of Lyapunov exponents

We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman’s result. Finally, we give a result about the growth of the spectral radius of products.

Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts

We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets

INEQUALITIES FOR NUMERICAL INVARIANTS OF SETS OF MATRICES

\Sigma

Continuity properties of the lower spectral radius

in

C 1 -generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

GL(d,\mathbb{R})

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, ℝ) cocycles

with the property that any cocycle with values in

Perturbation of the Lyapunov spectra of periodic orbits

\Sigma

Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms

has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a 2-dimensional result by Avila, Bochi, and Yoccoz. We give an example showing how these multicones can fail to have convexity properties.

Opening gaps in the spectrum of strictly ergodic Schrödinger operators

We prove three inequalities relating some invariants of sets of matrices, such as the joint spectral radius. One of the inequalities, in which proof we use geometric invariant theory, has the generalized spectral radius theorem of Berger and Wang as an immediate corollary.