Uniqueness of ergodic optimization of top Lyapunov exponent for typical matrix cocycles

The Entropy of Lyapunov-optimizing Measures of Some Matrix Cocycles

We consider one-step cocycles of 2ˆ 2 matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist, and are characterized by their support. Under an ...

The upper Lyapunov exponent of S1(2,R) cocycles: Discontinuity and the problem of positivity

Let T be an aperiodic automorphism of a standard probability space (X,m). Let V be the subset of A = L°°(X% 5/(2, R)) where the upper Lyapunov exponent is positive almost everywhere. We prove that the set V \ int(V) is not empty. So, there are always points in A where the Lyapunov exponents are discontinuous. We show further that the decision whether a given cocycle is in V is at least as hard ...

Ergodic Averages and Integrals of Cocycles

Abstract. This paper concerns the structure of the space C of real valued cocycles for a flow (X,Zm). We show that C is always larger than the set of cocycles cohomologous to the linear maps if the flow has a free dense orbit. By considering appropriate dual spaces for C, we obtain the concept of an invariant cocycle integral. The extreme points of the set of invariant cocycle integrals paralle...

Lyapunov Exponents For Some Quasi-Periodic Cocycles

We consider SL(2,R)-valued cocycles over rotations of the circle and prove that they are likely to have Lyapunov exponents ≈ ± logλ if the norms of all of the matrices are ≈ λ. This is proved for λ sufficiently large. The ubiquity of elliptic behavior is also observed. Consider an area preserving diffeomorphism f of a compact surface. Assume that f is not uniformly hyperbolic, but that it has o...

Lyapunov Exponent for Products of Random Matrices