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On the Aubry–Mather theory for symbolic dynamics

Published online by Cambridge University Press:  01 June 2008

E. GARIBALDI  and
A. O. LOPES
E. GARIBALDI
Affiliation:
Institut de Mathématiques, Université Bordeaux 1, F-33405 Talence, France (email: Eduardo.Garibaldi@math.u-bordeaux1.fr)
A. O. LOPES
Affiliation:
Instituto de Matemática, UFRGS, 91509-900 Porto Alegre, Brazil (email: artur.lopes@ufrgs.br)
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Abstract

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We propose a new model of ergodic optimization for expanding dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In other contexts (for twist maps, for instance), this property appears in a crucial way. A version of the Aubry–Mather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the two-sided shift. Under the transitive hypothesis, we show the existence of sub-actions for Hölder potentials also in the holonomic setting. We analyze then connections between calibrated sub-actions and the Mañé potential. A representation formula for calibrated sub-actions is presented, which drives us naturally to a classification theorem for these sub-actions. We also investigate properties of the support of maximizing probabilities.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 791 - 815
Copyright
Copyright © Cambridge University Press 2008

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