American Institute of Mathematical Sciences

August  2011, 30(3): 699-708. doi: 10.3934/dcds.2011.30.699

Equilibrium states of the pressure function for products of matrices

 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong 2 Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland

Received  April 2010 Revised  October 2010 Published  March 2011

Let $\{M_i\}_{i=1}^l$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1\cdots i_n\in \{1,\ldots, l\}^n$ such that $M_{i_1}\cdots M_{i_n}\ne$0. Let P : $(0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^l$. 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.
Citation: De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699
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