Equilibrium states of the pressure function for products of matrices
De-Jun Feng Antti Käenmäki
Discrete & Continuous Dynamical Systems - A 2011, 30(3): 699-708 doi: 10.3934/dcds.2011.30.699
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.
keywords: Thermodynamical formalism Products of matrices. Equilibrium states
The thermodynamic formalism for sub-additive potentials
Yongluo Cao De-Jun Feng Wen Huang
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 639-657 doi: 10.3934/dcds.2008.20.639
The topological pressure is defined for sub-additive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.
keywords: variational principle entropy topological pressure products of matrices. Thermodynamical formalism Lyapunov exponents

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