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I will present the problem of selection at temperature zero. Given a Dynamical System $X$ and $f o X$ and a potential $phi$ we look for the $f$-invariant measures $mu$ which maximize $int phi,dmu$. Such a measure is called a $phi$-maximizing measure. On the other hand, the Thermodynamical formalism produces an Equilibrium State $mu_eta$ for the potential $etaphi$ with $eta>0$. A ground state for $phi$ is a $phi$-maximizing measure which is an accumulation point for $mu_eta$ as $eta$ goes to $+infty$. The selection problem is to determine if $mu_eta$ converges, and also how it chooses the limit between all the $phi$-maximizing measures. The convergence does not always occur, but it turns out that fore some potentials, it is possible to prove convergence and to determine the limit. The talk will recall part of the background needed to understand and present the problem and will show some results. In particular, we emphasize that the Max-Plus formalism seems to be a powerful tool to solve the problem.
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