Antti Käenmäki: Domination and thermodynamic formalism for planar matrix cocycles

We consider cocycles in the simplest non-commutative setting, namely in the case of planar matrices. A cocycle is dominated if there is a uniform exponential gap between singular values of its iterates. This is equivalent to the existence of a strongly invariant multicone in the projective space. We show that a planar matrix cocycle is dominated if and only if matrices are hyperbolic and the norms in the generated sub-semigroup are almost multiplicative.
Matrix cocycles appear naturally in the study of random matrix products and in thermodynamic formalism for matrix-valued potentials. A norm potential satisfying domination is a prime example of an almost-additive dynamical system. We show that all such systems can be studied with the classical thermodynamic formalism. In fact, we are able to characterize all the properties of equilibrium states for norm potentials by means of the properties of matrices. As a consequence of our results, answering a folklore question, we show the existence of a quasi-Bernoulli equilibrium state which is not a Gibbs measure for any Hölder continuous potential.
The talk is based on a recent work with B. Bárány and I. D. Morris.