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The thermodynamic formalism for expanding maps

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Abstract

Letf:X↦X be an expanding map of a compact space (small distances are increased by a factor >1). A generating functionζ(z) is defined which countsf-periodic points with a weight. One can expressζ 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 ofζ(z). Gibbs distributions and applications to Julia sets are also discussed. Some new results are proved, and some natural conjectures are proposed.

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Authors and Affiliations

  1. I.H.E.S., F-91440, Bures-sur-Yvette, France

    David Ruelle

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  1. David Ruelle
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Communicated by J.-P. Eckmann

This is an expanded version of the Bowen lectures given by the author at U.C. Berkeley in November 1988

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Ruelle, D. The thermodynamic formalism for expanding maps. Commun.Math. Phys. 125, 239–262 (1989). https://doi.org/10.1007/BF01217908

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  • DOI: https://doi.org/10.1007/BF01217908

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