Thermodynamic formalism for null recurrent potentials |
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Authors: | Omri M. Sarig |
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Affiliation: | (1) School of Mathematical Sciences, Tel Aviv University Ramat Aviv, 69978 Tel Aviv, Israel |
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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(?) denote the Gurevic pressure of ? and letL ? 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 ? * ν=e P(?)ν,L ? h=e P(?) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL ? k , 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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