Weak KAM methods and ergodic optimal problems for countable Markov shifts

Details Weak KAM methods and ergodic optimal problems for countable Markov shifts Weak KAM methods and ergodic optimal problems for countable Markov shifts

2009-01-29

arxiv.org/abs/0901.4640v2

Mathematics

Dynamical Systems

15 pages. To appear in Bulletin of the Brazilian Mathematical Society.

Scientific paper

Let $\sigma:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ be the left shift acting on $ \boldsymbol{\Sigma} $, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $\sigma$-invariant Borel probabilities that maximize the integral of a given locally H\"older continuous potential $ A : \boldsymbol{\Sigma} \to \mathbb R $. 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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