On the Aubry–Mather theory for symbolic dynamics

By Eduardo Garibaldi, Artur Oscar Lopes. Ergodic Theory and Dynamical Systems, 2006

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Abstract 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1478258b4e0b88a145be01dec24ae5b249238aac