Equilibrium states of the pressure function for products of matrices

De-Jun Feng; Antti Käenmäki

doi:10.3934/dcds.2011.30.699

Article Contents

2011, Volume 30, Issue 3: 699-708. Doi: 10.3934/dcds.2011.30.699

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Equilibrium states of the pressure function for products of matrices

De-Jun Feng^1, and
Antti Käenmäki^2,

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

Received: April 2010

Revised: October 2010

Published: August 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D35; Secondary: 34D20.

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