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Thermodynamic formalism for null recurrent potentials. (English) Zbl 0992.37025

The paper deals with topological Markov shifts with countable number of states. The author recalls the definition of the Gurevich topological pressure of a Hölder continuous function \(\varphi\) and the corresponding Ruelle operator \(L_\varphi\). He classifies the function \( \varphi\) as recurrent, transient, positive recurrent or null recurrent according to the asymptotic behaviour of appropriate partition functions. He then proves that the Hölder continuous function \(\varphi\) is recurrent if and only if there exist \(\lambda>0\), a conservative measure \(\nu\), finite and positive on cylinders, and a positive continuous function \(h\) such that \(L^*_\varphi \nu= \lambda\nu\) and \(L_\varphi h=\lambda h\). Next he describes in detail the positive recurrent and the null recurrent case. He also studies thoroughly the corresponding dynamical \(\zeta\)-function and contributes to the theory of equilibrium states. This extends results in [the author, Ergodic Theory Dyn. Syst. 19, 1565-1593 (1999; Zbl 0994.37005)].

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.

Citations:

Zbl 0994.37005
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References:

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