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):  

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February 09, 2009

 

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