Flatness is a Criterion for Selection of Maximizing Measures

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.

References in corpus (4)

• Lyapunov spectrum of asymptotically sub-additive potentials
• On the zero-temperature limit of Gibbs states
• Chaotic temperature dependence at zero temperature
• Selection of measures for a potential with two maxima at the zero temperature limit

Cited by in corpus (7)

• Ergodic optimization in dynamical systems
• Ergodic optimization, zero temperature limits and the max-plus algebra
• Sensitive dependence of Gibbs measures at low temperatures
• Zero-temperature phase diagram for double-well type potentials in the summable variation class
• Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach
• On the selection of subaction and measure for perturbed potentials
• Poisson smooth structures on stratified symplectic spaces