# American Institute of Mathematical Sciences

October  2018, 38(10): 4997-5010. doi: 10.3934/dcds.2018218

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

 1 CIMAT, A.C., Jalisco S/N, Col. Valenciana CP: 36023 Guanajuato, Gto, Apartado Postal 402, CP 36000, México 2 Universidade Federal do Rio Grande do Sul, Instituto de Matemática, Av. Bento Gonçalves, 9500, Agronomia, Porto Alegre-RS, CEP: 91509-900 Caixa Postal: 15080, Brazil 3 Universidade Federal do Rio Grande do Sul, Campus Litoral Norte, Rod. RS 030, 11.700 Km 92, Emboaba, Tramandaí-RS, CEP: 95590-000, Brazil

* Corresponding author: Artur O. Lopes

Received  October 2017 Revised  April 2018 Published  July 2018

Fund Project: The second author is partially supported by CNPq

Consider $T(x) = d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $zhongwenzy<\lambda<1$ 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 Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A) = \sup_{\mu \in \mathcal{M}} \int A\, d\mu$.

For $\beta>0$ and $\lambda \in (0, 1)$, there exists a unique fixed point $u_{\lambda, \beta} :S^1\to \mathbb{R}$ 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 $\beta \to\infty$. Under these hypotheses we will show that $\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$ converges uniformly to the above $V$, as $\beta\to \infty$. 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 $\beta \to \infty$.

Citation: Renato Iturriaga, Artur O. Lopes, Jairo K. Mengue. Selection of calibrated subaction when temperature goes to zero in the discounted problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4997-5010. doi: 10.3934/dcds.2018218
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##### References:
 [1] Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383 [2] Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773 [3] Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics & Games, 2015, 2 (1) : 103-115. doi: 10.3934/jdg.2015.2.103 [4] Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675 [5] Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315 [6] Leandro Cioletti, Artur O. Lopes. Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6139-6152. doi: 10.3934/dcds.2017264 [7] Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623 [8] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [9] Yunmei Chen, Jiangli Shi, Murali Rao, Jin-Seop Lee. Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure. Inverse Problems & Imaging, 2015, 9 (1) : 79-103. doi: 10.3934/ipi.2015.9.79 [10] Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206 [11] Antoine Hochart. An accretive operator approach to ergodic zero-sum stochastic games. Journal of Dynamics & Games, 2019, 6 (1) : 27-51. doi: 10.3934/jdg.2019003 [12] François Delarue, Franco Flandoli. The transition point in the zero noise limit for a 1D Peano example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4071-4083. doi: 10.3934/dcds.2014.34.4071 [13] Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743 [14] Jiang Xu, Wen-An Yong. Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1319-1332. doi: 10.3934/dcds.2009.25.1319 [15] Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497 [16] Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806 [17] Toyohiko Aiki, Martijn Anthonissen, Adrian Muntean. On a one-dimensional shape-memory alloy model in its fast-temperature-activation limit. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 15-28. doi: 10.3934/dcdss.2012.5.15 [18] Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377 [19] Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027 [20] Aldana M. González Montoro, Ricardo Cao, Christel Faes, Geert Molenberghs, Nelson Espinosa, Javier Cudeiro, Jorge Mariño. Cross nearest-spike interval based method to measure synchrony dynamics. Mathematical Biosciences & Engineering, 2014, 11 (1) : 27-48. doi: 10.3934/mbe.2014.11.27

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