Some Examples of Nonuniformly Hyperbolic Cocycles

We consider some very simple examples of SL(2, R)-cocycles and prove that they have positive Lyapunov exponents. These cocycles form an open set in the C topology. Let f : (X,m) be a measure preserving transformation of a probability space, and let A : X → SL(2,R). With a slight abuse of language we call A an SL(2,R)-cocycle over the dynamical system (f,m). Let λ1 ≥ λ2 denote the Lyapunov exponents of (f,m;A). We are interested in whether or not (f,m;A) has nonzero Lyapunov exponents, or equivalently, whether or not λ1 > 0 a.e. Because norms of matrices are sub-multiplicative, the problem of estimating λ1 from below is in general a rather difficult one. This note is an attempt to add to the existing pool of techniques for proving positive exponents. We consider some very simple cocycles defined over z 7→ z , z ∈ S, or automorphisms of the 2-torus, and give a positive lower bound for λ1. These examples can be made C r for any r ≤ ω. If C(X,SL(2,R)) denotes the space of C maps from S or T to SL(2,R) endowed with the C topology, then our examples fill up an open set in C(X,SL(2,R)) – although none of them is uniformly hyperbolic. This openness part of our assertion should probably be contrasted with a theorem of Mañé [M], in which he proves that away from Anosov components, the generic C area-preserving diffeomorphism of a compact 2-dimensional surface has zero exponents a.e. Since x 7→ Dfx is a C cocycle if f is C, Mañé’s methods suggest that in our setting, the set of non-uniformly hyperbolic A’s form a first category set in C(X,SL(2,R)). This research is partially supported by NSF.