Equilibrium states of the pressure function for products of matrices

By De-Jun Feng, A. Kaenmaki. 2010

67 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8a5fca32cd12f126ac0e1dda8535606cfe4f1a52

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Dimension of invariant measures for affine iterated function systems (2019, unknown): “1 in [29].”
Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles (2013, background): “The function M(A, ·) is differentiable whenever there is just one equilibrium measure, and this is known to hold under several conditions, such as irreducibility or strict positivity, see [23, 22].”