The entropy of Lyapunov-optimizing measures of some matrix cocycles
Jairo Bochi - Facultad deMatemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile (email) 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.
Keywords: Joint spectral radius, joint spectral subradius, ergodic optimization, Lyapunov exponents, linear cocycles, dominated splittings, entropy.
Received: April 2015; Revised: April 2016; Available Online: July 2016. |