Flatness is a Criterion for Selection of Maximizing Measures
Dynamical Systems
2015-06-03 v1
Abstract
For a full shift with Np+1 symbols and for a non-positive potential, locally proportional to the distance to one of N disjoint full shifts with p symbols, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest. In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes. As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction.
Cite
@article{arxiv.1201.5452,
title = {Flatness is a Criterion for Selection of Maximizing Measures},
author = {Renaud Leplaideur},
journal= {arXiv preprint arXiv:1201.5452},
year = {2015}
}