THERMOGAMAS - Dichotomy Theorem for Minimizing Configurations and Minimizing Measures in the Discrete Aubry-Mather Model (Gaël Meignan)

Talk given during the THERMOGAMAS Workshop.

Abstract: We present results over existence and characterization of minimizing configurations and minimizing measures on the one dimensional lattice ℤ. In order to generalize what is known for nearest neighbor interactions to finite range interactions, we first provide dichotomy results over the set of minimizing configurations. We prove for φ∈ℝ^ℤ minimizing configuration the equivalence between several conditions.  

Then, we discuss the existence and characterization of invariant minimizing measures defined on the space of configurations. In this direction, we give a characterization of the ground state energy of an interaction. This will allow us to assert that the support of such minimizing measures is included in the minimizing configurations of our model. Finally, thanks to our first dichotomy result, these supports are included in configurations with strong rotation number or even satisfying the Birkhoff property depending on the interaction.