Abstract
This is an article I wrote for Dynamics, Games, and Science. In Dynamics, Game, and Science, one of the most important equilibrium states is a Gibbs state. The deformation of a Gibbs state becomes an important subject in these areas. An appropriate metric on the space of underlying dynamical systems is going to be very helpful in the study of deformation. The Teichmüller metric becomes a natural choice. The Teichmüller metric, just like the hyperbolic metric on the open unit disk, makes the space of underlying dynamical systems a complete space. The Teichmüller metric precisely measures the change of the eigenvalues at all periodic points which are essential data needed to obtain the Gibbs state for a given dynamical system. In this article, I will introduce the Teichmüller metric and, subsequently, a generalization of Gibbs theory which we call geometric Gibbs theory.
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Acknowledgements
I would like to thank my student John Adamski who read the initial version of this manuscript very carefully and found many typos and made very good suggestions to improve the exposition of this paper. This research is partially supported by the collaboration grant (#199837) from the Simons Foundation, the CUNY collaborative incentive research grants (#1861 and #2013), and awards from PSC-CUNY. This research is also partially supported by the collaboration grant (#11171121) from the NSF of China and a collaboration grant from Academy of Mathematics and Systems Science and the Morningside Center of Mathematics at the Chinese Academy of Sciences.
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Jiang, Y. (2015). An Introduction to Geometric Gibbs Theory. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Dynamics, Games and Science. CIM Series in Mathematical Sciences, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16118-1_18
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