Almost Reduction and Perturbation of Matrix Cocycles

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles for general matrix cocycles. 1 From zero Lyapunov exponents to rotation cocycles 1.1 Basic definitions Let F : Ω Ñ Ω be a homeomorphism of a compact metric space Ω. Let V be a finite-dimensional real vector bundle over Ω, whose fiber over ω is denoted by Vω. Let A be a vector-bundle automorphism that fibers over F ; this means that the restriction of A to each fiber Vω is a linear automorphism Apωq onto VFω. In the case of trivial vector bundles, A is usually called a linear cocycle. As a convention, automorphisms of V will be denoted by calligraphic letters, and the restrictions to the fibers will be denoted by the corresponding roman letters. Analogously, for any integer n, the restriction of the power A to the fiber Vω is denoted by A pωq; thus Apωq “ ApFn ́1ωq ̋ ̈ ̈ ̈ ̋Apωq for n ą 0. A Riemannian metric on V is a continuous choice of inner product x ̈, ̈yω on each fiber Vω. It induces a Riemannian norm }v}ω “ a xv, vyω . Given a linear map L : Vω Ñ Vω1 , its norm }L} and its conorm mpLq are defined respectively as the supremum and the infimum of }Lv}ω1 over all unit vectors v P Vω. Let AutpV, F q denote the space of all automorphisms of V that fiber over F , endowed with the topology induced by the distance dpA,Bq “ supω }Apωq ́ Bpωq}, for some choice of a Riemannian norm on V . ̊Partially supported by CNPq (Brazil) and FAPERJ (Brazil). :Partially supported by the Fondecyt Project 1120131 and the “Center of Dynamical Systems and Related Topics” (DySyRF; ACT-Project 1103, Conicyt). 1.2 Uniform subexponential growth and its consequences