## Relatório de pesquisa 05/09

Weak
KAM Methods and Ergodic Optimal Problems for Countable Markov Shifts,
Rodrigo Bissacot and Eduardo Garibaldi, submitted February 09, 2009.

Abstract

Let $\sigma\colon \Sigma\to \Sigma$ be the left shift acting on
$\Sigma$, 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 $A\colon \Sigma\to \mathbb{R}$. 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).

Mathematics Subject Classifications
(2000):

Keywords:

February 09, 2009

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