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

2018
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Discrete and Continuous Dynamical Systems. Series A
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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

doi:10.3934/dcds.2018218
fatcat:6vni2lqgqzbgtarhyamipg24yi