On the Aubry-Mather theory for symbolic dynamics

• Autor(a) principal: Garibaldi, Eduardo
• Data de Publicação: 2008
• Outros Autores: Lopes, Artur Oscar
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFRGS
• Texto Completo: http://hdl.handle.net/10183/27440
• topic: Otimização ergódica; Sistemas dinamicos : Ergodicidade : Topologia

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 Holder potentials also in the holonomic setting. We analyze then connections between calibrated sub-actions and the Ma˜n´e 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.