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Thermodynamic formalism for null recurrent potentials

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Abstract

We extend Ruelle’s Perron-Frobenius theorem to the case of Hölder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(ϕ) denote the Gurevic pressure of ϕ and letL ϕ be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatL *ϕ ν=e P(ϕ)ν,L ϕ h=e P(ϕ) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL kϕ , and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.

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  1. School of Mathematical Sciences, Tel Aviv University Ramat Aviv, 69978, Tel Aviv, Israel

    Omri M. Sarig

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  1. Omri M. Sarig
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Sarig, O.M. Thermodynamic formalism for null recurrent potentials. Isr. J. Math. 121, 285–311 (2001). https://doi.org/10.1007/BF02802508

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  • DOI: https://doi.org/10.1007/BF02802508

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