2017/10/16 by Iturriaga, Renato, Lopes, Artur O., Mengue, Jairo K.
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.1710.05974
Consider T(x)= d x (mod 1) acting on S1, a Lipschitz potential A:S1 → ℝ, 00 and λ∈ (0,1), there exists a unique fixed point uλ,β :S1→ ℝ for the equation e^uλ,β(x) = ∑T(y)=xe^βA(y) +λuλ,β(y). It is known that as λ→ 1 the family e^[uλ,β- sup uλ,β] converges uniformly to the main eigenfuntion ϕβ for the Ruelle operator associated to βA. We consider λ=λ(β), β(1-λ(β))→+∞ and λ(β) → 1, as β→∞. Under these hypothesis we will show that \frac1β(uλ,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→ ∞.