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- W4300785906 abstract "Consider $T(x)= d , x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 to mathbb{R}$, $0<lambda<1$ and the unique function $b_lambda:S^1 to mathbb{R}$ satisfying $ b_lambda(x) = max_{T(y)=x} { lambda , b_lambda(y) + A(y)}.$ We will show that, when $lambda to 1$, the function $b_lambda- frac{m(A)}{1-lambda}$ converges uniformly to the calibrated subaction $V(x) = max_{mu in mathcal{ M}} int S(y,x) , d mu(y)$, where $S$ is the Ma~ne potential, $mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)= sup_{mu in mathcal{M}} int A,dmu$. For $beta>0$ and $lambda in (0,1)$, there exists a unique fixed point $u_{lambda,beta} :S^1to mathbb{R}$ for the equation $e^{u_{lambda,beta}(x)} = sum_{T(y)=x}e^{beta A(y) +lambda u_{lambda,beta}(y)}$. It is known that as $lambda to 1$ the family $e^{[u_{lambda,beta}- sup u_{lambda,beta}]}$ converges uniformly to the main eigenfuntion $phi_beta $ for the Ruelle operator associated to $beta A$. We consider $lambda=lambda(beta)$, $beta(1-lambda(beta))to+infty$ and $lambda(beta) to 1$, as $beta toinfty$. Under these hypothesis we will show that $frac{1}{beta}(u_{lambda,beta}-frac{P(beta A)}{1-lambda})$ converges uniformly to the above $V$, as $betato infty$. The parameter $beta$ represents the inverse of temperature in Statistical Mechanics and $beta to infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $beta to infty$." @default.
- W4300785906 created "2022-10-04" @default.
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- W4300785906 date "2017-10-16" @default.
- W4300785906 modified "2023-09-26" @default.
- W4300785906 title "Selection of calibrated subaction when temperature goes to zero in the discounted problem" @default.
- W4300785906 doi "https://doi.org/10.48550/arxiv.1710.05974" @default.
- W4300785906 hasPublicationYear "2017" @default.
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