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Geometric Gibbs theory. (English) Zbl 1453.37029

The theory of Gibbs measures as used in the study of Sinai-Ruelle-Bowen (SRB) measures in hyperbolic dynamical systems leads in particular to the fundamental concept of an equilibrium state. One direction of enquiry is to study deformations of a Gibbs measure and its associated density, using an appropriate metric on the space of Gibbs measures. Here a complex Banach manifold structure on the space of Gibbs measures is constructed, and a generalised geometric Gibbs measure is introduced and studied. This extends the familiar Gibbs theory from smooth potentials to continuous ones; existence, uniqueness, and compatibility of these geometric Gibbs measures are studied, and they are shown to be equilibrium states.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37E10 Dynamical systems involving maps of the circle
37A05 Dynamical aspects of measure-preserving transformations
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