Weak KAM methods and ergodic optimal problems for countable Markov shifts |
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Authors: | Rodrigo Bissacot Eduardo Garibaldi |
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Institution: | 1. Departamento Matemática, UFMG, 30161-970, Belo Horizonte, MG, Brazil 2. Departamento de Matemática, UNICAMP, 13083-859, Campinas, SP, Brazil
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Abstract: | Let σ: Σ → Σ be the left shift acting on Σ, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of σ-invariant Borel probabilities that maximize the integral of a given locally Hölder continuous potential A: Σ → ?. Under certain conditions, we are able to show not only that A-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions). |
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