Equilibrium states of the pressure function for products of matrices

Abstract

Let $\{M_i{\}}_{i=1}^{\ell }$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \mathbb{N}$, there exists $i_1...i_n\in \{1,...,\ell {\}}^n$ such that $M_{i_1}...M_{i_n}\ne 0$. Let $P:(0,\infty )\to \mathbb{R}$ 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.

12 pages. To appear in DCDS

References in corpus (3)

• Lyapunov spectrum of asymptotically sub-additive potentials
• Dimension and measures on sub-self-affine sets
• Weighted equilibrium states for factor maps between subshifts

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