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DCDS

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.
DCDS

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.

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