The Entropy of Lyapunov-optimizing Measures of Some Matrix Cocycles
نویسنده
چکیده
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 additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunovoptimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step SLp2,Rq-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.
منابع مشابه
Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics
Introduction 1 1. Lyapunov exponents of dynamical systems 3 2. Examples of systems with nonzero exponents 6 3. Lyapunov exponents associated with sequences of matrices 18 4. Cocycles and Lyapunov exponents 24 5. Regularity and Multiplicative Ergodic Theorem 31 6. Cocycles over smooth dynamical systems 46 7. Methods for estimating exponents 54 8. Local manifold theory 62 9. Global manifold theor...
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