Journal of Modern Dynamics (JMD)

The entropy of Lyapunov-optimizing measures of some matrix cocycles

Pages: 255 - 286, Volume 10, 2016      doi:10.3934/jmd.2016.10.255

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Jairo Bochi - Facultad deMatemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile (email)
Michal Rams - Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8. 00-956 Warsaw, Poland (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.
Mathematics Subject Classification:  Primary: 15B48; Secondary: 37H15, 37D30, 93C30.

Received: April 2015;      Revised: April 2016;      Available Online: July 2016.