Geometric Gibbs theory
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https://doi.org/10.1007/s11425-019-1638-6
Geometric Gibbs theory In Memory of Professor Shantao Liao
Yunping Jiang1,2 1Department
of Mathematics, Queens College of the City University of New York, New York, NY 11367-1597, USA; 2Department of Mathematics, The CUNY Graduate Center, New York, NY 10016, USA Email: [email protected] Received July 19, 2019; accepted November 19, 2019
Abstract
We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain
continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero. Keywords
geometric Gibbs measure, continuous potential, smooth potential, Teichm¨ uller’s metric, maximum
metric, Kobayashi’s metric, symmetric rigidity, complex Banach manifold MSC(2010)
37D35, 37F30, 37E10, 37A05
Citation: Jiang Y P. Geometric Gibbs theory. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-0191638-6
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Introduction
The mathematical theory of Gibbs measures, an important idea originating in physics, is a beautiful mathematical theory from the celebrated work of Sinai [28, 29] and Ruelle [25, 26]. It leads to the study of Sinai-Ruelle-Bowen (SRB)-measures in Anosov dynamical systems and, more generally, Axiom A dynamical systems due to the further work of Sinai, Ruelle, Bowen and many others (see [2]). An essential feature of a Gibbs measure is that it is an equilibrium state. This equilibrium state plays a significant role in mathematics, as well as many other areas such as physics, chemistry, biology, economics, and game theory. An important topic in the current study of Gibbs measures is to study the deformation of a Gibbs measure and its density (see, for example, [20, 27]). To have a nice deformation theory, an appropriate metric on the space of all Gibbs measures is helpful. In this paper, we introduce a complex Banach manifold structure on the space of all Gibbs measures and, then study a generalized Gibbs measure which we call a geometric Gibbs measure. A classical Gibbs measure is for a smooth potential (i.e., at least H¨older smoothness). In general, for a continuous potential, one may not have an appropriate Gibbs theory. We prove that for a specific c Science China Press and Springer-Verlag GmbH Germany, part of Springer N
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