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Existence of Gibbs measures for countable Markov shifts. (English) Zbl 1009.37003

Summary: We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of R. D. Mauldin and M. Urbánski [Isr. J. Math. 125, 93-130 (2001; Zbl 1016.37005)] who showed that this condition is sufficient.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37B10 Symbolic dynamics

Citations:

Zbl 1016.37005
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References:

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