The entropy of Lyapunov-optimizing measures of some matrix cocycles

TL;DR

This paper investigates the properties of Lyapunov-optimizing measures for certain matrix cocycles, showing that under domination conditions these measures have zero entropy, contrasting with non-dominated cases where positive entropy measures can exist.

Contribution

The paper proves that Lyapunov-optimizing measures for dominated cocycles have zero entropy, providing a new characterization under specific cone nonoverlapping conditions.

Findings

Lyapunov-optimizing measures exist and are characterized by their support in dominated cocycles.

Under cone nonoverlapping conditions, these measures have zero entropy.

Without domination, measures of positive entropy with zero Lyapunov exponent can occur.

Abstract

We consider one-step cocycles of $2 \times 2$ 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 Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.