Selection of calibrated subaction when temperature goes to zero in the discounted problem

Renato Iturriaga, Artur O. Lopes, Jairo K. Mengue
2018 Discrete and Continuous Dynamical Systems. Series A  
Consider T (x) = d x (mod 1) acting on S 1 , a Lipschitz potential A : S 1 → R, 0 < λ < 1 and the unique function b λ : We will show that, when λ → 1, the function b λ − m(A) 1−λ converges uniformly to the calibrated subaction V (x) = max µ∈M S(y, x) dµ(y), where S is the Mañe potential, M is the set of invariant probabilities with support on the Aubry set and m(A) = sup µ∈M A dµ. For β > 0 and λ ∈ (0, 1), there exists a unique fixed point u λ,β : S 1 → R for the equation e u λ,β (x) = T (y)=x
more » ... λ,β (x) = T (y)=x e βA(y)+λu λ,β (y) . It is known that as λ → 1 the family e [u λ,β −sup u λ,β ] converges uniformly to the main eigenfuntion φ β for the Ruelle operator associated to βA. We consider λ = λ(β), β(1 − λ(β)) → +∞ and λ(β) → 1, as β → ∞. Under these hypotheses we will show that 1 β (u λ,β − P (βA) 1−λ ) converges uniformly to the above V , as β → ∞. The parameter β represents the inverse of temperature in Statistical Mechanics and β → ∞ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when β → ∞. 2010 Mathematics Subject Classification. Primary: 37D35. Key words and phrases. Calibrated subaction, maximizing measure, discounted method, zero temperature limit, eigenfunction of the Ruelle operator. The second author is partially supported by CNPq.
doi:10.3934/dcds.2018218 fatcat:6vni2lqgqzbgtarhyamipg24yi