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 and Continuous Dynamical Systems, 2018, 38 (10) : 4997-5010. doi: 10.3934/dcds.2018218
References:
 [1] R. Bamon, J. Kiwi, J. Rivera-Letelier and R. Urzua, On the topology of solenoidal attractors of the cylinder, Ann. Inst. H. Poincaré Anal. Non Lineaire, 23 (2006), 209-236.  doi: 10.1016/j.anihpc.2005.03.002. [2] A. Baraviera, R. Leplaideur and A. O. Lopes, Ergodic Optimization, Zero Temperature and the Max-Plus Algebra, 29° Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, 2013. [3] A. T. Baraviera, L. M. Cioletti, A. Lopes, J. Mohr and R. R. Souza, On the general one-dimensional XY Model: Positive and zero temperature, selection and non-selection, Reviews in Math. Physics., 23 (2011), 1063-1113.  doi: 10.1142/S0129055X11004527. [4] T. Bousch, La condition de Walters, Ann. Sci. ENS, 34 (2001), 287-311.  doi: 10.1016/S0012-9593(00)01062-4. [5] J. Bremont, Gibbs measures at temperature zero, Nonlinearity, 16 (2003), 419-426.  doi: 10.1088/0951-7715/16/2/303. [6] G. Contreras, A. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory and Dynamical Systems, 21 (2001), 1379-1409.  doi: 10.1017/S0143385701001663. [7] G. Contreras, A. O. Lopes and Ph. Thieullen, Maximizing measures for expanding transformations, preprint, arXiv:1307.0533. [8] G. Contreras, A. O. Lopes and E. Oliveira, Ergodic Transport Theory, periodic maximizing probabilities and the twist condition, in Modeling, Optimization, Dynamics and Bioeconomy I (eds. D. Zilberman and A. Pinto), Springer Proceedings in Mathematics and Statistics, vol. 73, Edit., (2014), 183-219. doi: 10.1007/978-3-319-04849-9_12. [9] J. P. Conze and Y. Guivarch, Croissance Des Sommes Ergodiques, manuscript, circa, 1993. [10] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6. [11] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted equation: The discrete case, Math. Z., 284 (2016), 1021-1034.  doi: 10.1007/s00209-016-1685-y. [12] E. Garibaldi and A. O. Lopes, On the Aubry-Mather theory for symbolic dynamics, Erg. Theo. and Dyn. Systems, 28 (2008), 791-815.  doi: 10.1017/S0143385707000491. [13] E. Garibaldi, Ergodic Optimization in the Expanding Case: Concepts, Tools and Applications, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-66643-3. [14] E. Garibaldi and Ph. Thieullen, Description of some ground states by Puiseux techniques, Journal of Statistical Physics, 146 (2012), 125-180.  doi: 10.1007/s10955-011-0357-x. [15] O. Jenkinson, Ergodic optimization, Discrete and Continuous Dynamical Systems, Series A, 15 (2006), 197-224.  doi: 10.3934/dcds.2006.15.197. [16] R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18 (2005), 2847-2880.  doi: 10.1088/0951-7715/18/6/023. [17] A. O. Lopes and E. R. Oliveira, On the thin boundary of the fat attractor, in Modeling, Dynamics, Optimization and Bioeconomics III (eds. A. Pinto and D. Zilberman), Springer Proceedings in Mathematics and Statistics, Springer Verlag, (2018), 205-246. [18] A. O. Lopes, J. K. Mengue, J. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a-priori probability: positive and zero temperature, Erg. Theory and Dyn. Systems, 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15. [19] A. O. Lopes, Thermodynamic Formalism, Maximizing Probabilities and Large Deviations, Work in progress. Lecture Notes - Dynamique en Cornouaille, 2012. [20] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310.  doi: 10.1088/0951-7715/9/2/002. [21] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 1-268. [22] M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027.  doi: 10.1088/0951-7715/14/5/306.

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References:
 [1] R. Bamon, J. Kiwi, J. Rivera-Letelier and R. Urzua, On the topology of solenoidal attractors of the cylinder, Ann. Inst. H. Poincaré Anal. Non Lineaire, 23 (2006), 209-236.  doi: 10.1016/j.anihpc.2005.03.002. [2] A. Baraviera, R. Leplaideur and A. O. Lopes, Ergodic Optimization, Zero Temperature and the Max-Plus Algebra, 29° Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, 2013. [3] A. T. Baraviera, L. M. Cioletti, A. Lopes, J. Mohr and R. R. Souza, On the general one-dimensional XY Model: Positive and zero temperature, selection and non-selection, Reviews in Math. Physics., 23 (2011), 1063-1113.  doi: 10.1142/S0129055X11004527. [4] T. Bousch, La condition de Walters, Ann. Sci. ENS, 34 (2001), 287-311.  doi: 10.1016/S0012-9593(00)01062-4. [5] J. Bremont, Gibbs measures at temperature zero, Nonlinearity, 16 (2003), 419-426.  doi: 10.1088/0951-7715/16/2/303. [6] G. Contreras, A. O. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory and Dynamical Systems, 21 (2001), 1379-1409.  doi: 10.1017/S0143385701001663. [7] G. Contreras, A. O. Lopes and Ph. Thieullen, Maximizing measures for expanding transformations, preprint, arXiv:1307.0533. [8] G. Contreras, A. O. Lopes and E. Oliveira, Ergodic Transport Theory, periodic maximizing probabilities and the twist condition, in Modeling, Optimization, Dynamics and Bioeconomy I (eds. D. Zilberman and A. Pinto), Springer Proceedings in Mathematics and Statistics, vol. 73, Edit., (2014), 183-219. doi: 10.1007/978-3-319-04849-9_12. [9] J. P. Conze and Y. Guivarch, Croissance Des Sommes Ergodiques, manuscript, circa, 1993. [10] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions, Invent. Math., 206 (2016), 29-55.  doi: 10.1007/s00222-016-0648-6. [11] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted equation: The discrete case, Math. Z., 284 (2016), 1021-1034.  doi: 10.1007/s00209-016-1685-y. [12] E. Garibaldi and A. O. Lopes, On the Aubry-Mather theory for symbolic dynamics, Erg. Theo. and Dyn. Systems, 28 (2008), 791-815.  doi: 10.1017/S0143385707000491. [13] E. Garibaldi, Ergodic Optimization in the Expanding Case: Concepts, Tools and Applications, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-66643-3. [14] E. Garibaldi and Ph. Thieullen, Description of some ground states by Puiseux techniques, Journal of Statistical Physics, 146 (2012), 125-180.  doi: 10.1007/s10955-011-0357-x. [15] O. Jenkinson, Ergodic optimization, Discrete and Continuous Dynamical Systems, Series A, 15 (2006), 197-224.  doi: 10.3934/dcds.2006.15.197. [16] R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18 (2005), 2847-2880.  doi: 10.1088/0951-7715/18/6/023. [17] A. O. Lopes and E. R. Oliveira, On the thin boundary of the fat attractor, in Modeling, Dynamics, Optimization and Bioeconomics III (eds. A. Pinto and D. Zilberman), Springer Proceedings in Mathematics and Statistics, Springer Verlag, (2018), 205-246. [18] A. O. Lopes, J. K. Mengue, J. Mohr and R. R. Souza, Entropy and variational principle for one-dimensional lattice systems with a general a-priori probability: positive and zero temperature, Erg. Theory and Dyn. Systems, 35 (2015), 1925-1961.  doi: 10.1017/etds.2014.15. [19] A. O. Lopes, Thermodynamic Formalism, Maximizing Probabilities and Large Deviations, Work in progress. Lecture Notes - Dynamique en Cornouaille, 2012. [20] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310.  doi: 10.1088/0951-7715/9/2/002. [21] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187/188 (1990), 1-268. [22] M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027.  doi: 10.1088/0951-7715/14/5/306.
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