Equilibrium states of the pressure function for products of matrices

Details Equilibrium states of the pressure function for products of matrices Equilibrium states of the pressure function for products of matrices

2010-09-16

arxiv.org/abs/1009.3129v1

Mathematics

Dynamical Systems

12 pages. To appear in DCDS

Scientific paper

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.

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