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Thermodynamic formalism for countable to one Markov systems. (English) Zbl 1095.37007
The author defined Markov systems with finite range structure in [Tokyo J. Math. 9, 457–485 (1986; Zbl 0625.58001] which was later extended to Markov fibred systems by [J. Aaronson, M. Denker and M. Urbanski [Trans. Am. Math. Soc. 337, No. 2, 495–548 (1993; Zbl 0789.28010)]. The thermodynamic formalism in the sense of the Bowen-Ruelle program has been developed in recent years. Here, the author makes another essential step to complete the theory.
The starting point are potentials \(\phi\) of weak bounded variation. It is possible to define a notion of topological pressure for these potentials via the canonical sequence space representation (which is noncompact). This notion shares many of the properties known for the standard pressure function. Next, noninvariant Gibbs measures are constructed (in case the Schweiger property and the property of local exponential instability hold). The author also provides a discussion (for maps with parabolic points) when Gibbs measures do not exist. Another result shows the existence of an invariant, equivalent measures which maximize \(h_\mu(T)+ \int\phi\,d\mu\) in the variational formula. The set of equilibrium states is characterized. In case of a potential with local bounded distortion, a Gibbs measure is constructed which is not necessarily finite.

MSC:
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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