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The thermodynamic formalism for expanding maps. (English) Zbl 0702.58056
Summary: Let f: \(X\mapsto X\) be an expanding map of a compact space (small distances are increased by a factor \(>1)\). A generating function \(\zeta\) (z) is defined which counts f-periodic points with a weight. One can express \(\zeta\) in terms of nonstandard “Fredholm determinants” of certain “transfer operators”, which can be studied by methods borrowed from statistical mechanics. In this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of \(\zeta\) (z). Gibbs distributions and applications to Julia sets are also discussed. Some new results are proved, and some natural conjectures are proposed.

MSC:
37D99 Dynamical systems with hyperbolic behavior
82B30 Statistical thermodynamics
37E99 Low-dimensional dynamical systems
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