Let $\{ M_i\}_{ i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{ 1,..., \ell\}^n$ such that $M_{ i_1}... M_{ i_n}\neq { \bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{ M_i\}_{ i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.
Categories: math.DS math.CA