# American Institute of Mathematical Sciences

July  2011, 29(3): 1155-1174. doi: 10.3934/dcds.2011.29.1155

## Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics

 1 Instituto de Matemática, UFRGS, 91509-900, Porto Alegre, Brazil 2 Departamento de Matemática, PUC, 22453-900, Rio de Janeiro, RJ, Brazil

Received  February 2010 Revised  June 2010 Published  November 2010

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_{\lambda}(p,v)=\frac{1}{2}g_{p}(v,v)- \lambda\omega_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $\omega$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-\omega_{p}(v)$ has support in a unique closed geodesic $\gamma$; and the curvature is negative at every point of $M$ but at the points of $\gamma$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $\gamma$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.
Citation: Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
##### References:
 [1] N. Anantharaman, On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics,, J. Eur. Math. Soc., 6 (2004), 207.  doi: 10.4171/JEMS/9.  Google Scholar [2] N. Anantharaman, Counting geodesics which are optimal in homology,, Erg. Theo. and Dyn. Syst., 23 (2003), 353.  doi: 10.1017/S0143385702001372.  Google Scholar [3] N. Anantharaman, Entropie et localisation des fonctions propres,, ENS-Lyon, (2006).   Google Scholar [4] N. Anantharaman, R. Iturriaga, P. Padilla and H. Sánchez-Morgado, Physical solutions of the Hamilton-Jacobi equation,, Disc. Contin. Dyn. Syst. Ser. B, 5 (2005), 513.  doi: 10.3934/dcdsb.2005.5.513.  Google Scholar [5] W. Ballman, M. Gromov and V. Schroeder, "Manifolds of Non-positive Curvature,", Birkhausser, (1985).   Google Scholar [6] P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation,, Math. Res. Lett., 14 (2007), 503.   Google Scholar [7] H. Busemann, "The Geometry of Geodesics,", Academic Press, (1955).   Google Scholar [8] M. J. D. Carneiro, On minimizing measures of the action of autonomous Lagrangians,, Nonlinearity, 8 (1995), 1077.  doi: 10.1088/0951-7715/8/6/011.  Google Scholar [9] J. Cheeger and D. G. Ebin, "Comparison Theorems in Riemannian Geometry,", North-Holland Mathematical Library, 9 (1975).   Google Scholar [10] G. Contreras, Action potential and weak KAM solutions,, Calc. Var. Partial Differential Equations, 13 (2001), 427.   Google Scholar [11] G. Contreras and R. Ruggiero, Non-hyperbolic surfaces having all ideal triangles of finite area,, Bul. Braz. Math. Soc., 28 (1997), 43.  doi: 10.1007/BF01235988.  Google Scholar [12] G. Contreras and R. Iturriaga, Global minimizers of autonomous lagrangians,, to appear, (2004).   Google Scholar [13] G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values,, Geom. Funct. Anal., 8 (1998), 788.  doi: 10.1007/s000390050074.  Google Scholar [14] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Springer Verlag, (1998).   Google Scholar [15] P. Eberlein, Manifolds of nonpositive curvature,, in, 27 (1989), 223.   Google Scholar [16] R. Durrett, "Stochastic Calculus: A practical Introduction,", CRC-Press, (1996).   Google Scholar [17] L. C. Evans, Towards a quantum analog of weak KAM theory,, Commun. in Math. Phys., 244 (2004), 311.  doi: 10.1007/s00220-003-0975-5.  Google Scholar [18] A. Fathi, Weak KAM theorem and lagrangian dynamics,, to appear, (2004).   Google Scholar [19] A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls,, C. R. Acad. Sci. Paris Ser. I, 235 (1997), 649.   Google Scholar [20] A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation,, Invent. Math., 155 (2004), 363.  doi: 10.1007/s00222-003-0323-6.  Google Scholar [21] M. Gerber and V. Nitica, Holder exponents of horocycle foliations on surfaces,, Ergodic Theory Dynam. Systems, 19 (1999), 1247.  doi: 10.1017/S0143385799146832.  Google Scholar [22] D. A. Gomes and E. Valdinoci, Entropy penalization methods for Hamilton-Jacobi equations,, Adv. Math., 215 (2007), 94.  doi: 10.1016/j.aim.2007.04.001.  Google Scholar [23] D. A. Gomes, A. O. Lopes and J. Mohr, The Mather measure and a large deviation principle for the entropy penalized method,, to appear in Communications in Contemporary Mathematics., ().   Google Scholar [24] D. A. Gomes, A. O. Lopes and J. Mohr, Wigner measures and the semi-classical limit to the Aubry-Mather measure,, to appear, (2009).   Google Scholar [25] A. O. Lopes, V. Rosas and R. Ruggiero, Cohomology and Subcohomology for expansive geodesic flows,, Discrete and Continous Dynamical Systems, 17 (2007), 403.   Google Scholar [26] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems,, Nonlinearity, 9 (1996), 273.  doi: 10.1088/0951-7715/9/2/002.  Google Scholar [27] R. Markarian, Billiards with polynomial decay of correlations,, Ergodic Theory Dynam. Systems, 24 (2004), 177.  doi: 10.1017/S0143385703000270.  Google Scholar [28] D. Massart, Aubry sets vs Mather sets in two degrees of freedom,, to appear, (2008).   Google Scholar [29] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 2 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar [30] S. Nonnenmacher, Some open questions in wave chaos,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/8/T01.  Google Scholar [31] R. Ruggiero., Expansive geodesic flows in manifolds with no conjugate points,, Ergodic Theory Dynam. Systems, 17 (1997), 211.  doi: 10.1017/S0143385797060963.  Google Scholar [32] R. Ruggiero, Expansive dynamics and hyperbolic geometry,, Bul. Braz. Math. Soc., 25 (1994), 139.  doi: 10.1007/BF01321305.  Google Scholar [33] O. M. Sarig., Subexponential decay of correlations,, Invent. Math., 150 (2002), 629.  doi: 10.1007/s00222-002-0248-5.  Google Scholar [34] D. Stroock, "Partial Differential Equations for Probabilists,", Cambridge Press, (2008).  doi: 10.1017/CBO9780511755255.  Google Scholar [35] L. S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153.  doi: 10.1007/BF02808180.  Google Scholar

show all references

##### References:
 [1] N. Anantharaman, On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics,, J. Eur. Math. Soc., 6 (2004), 207.  doi: 10.4171/JEMS/9.  Google Scholar [2] N. Anantharaman, Counting geodesics which are optimal in homology,, Erg. Theo. and Dyn. Syst., 23 (2003), 353.  doi: 10.1017/S0143385702001372.  Google Scholar [3] N. Anantharaman, Entropie et localisation des fonctions propres,, ENS-Lyon, (2006).   Google Scholar [4] N. Anantharaman, R. Iturriaga, P. Padilla and H. Sánchez-Morgado, Physical solutions of the Hamilton-Jacobi equation,, Disc. Contin. Dyn. Syst. Ser. B, 5 (2005), 513.  doi: 10.3934/dcdsb.2005.5.513.  Google Scholar [5] W. Ballman, M. Gromov and V. Schroeder, "Manifolds of Non-positive Curvature,", Birkhausser, (1985).   Google Scholar [6] P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation,, Math. Res. Lett., 14 (2007), 503.   Google Scholar [7] H. Busemann, "The Geometry of Geodesics,", Academic Press, (1955).   Google Scholar [8] M. J. D. Carneiro, On minimizing measures of the action of autonomous Lagrangians,, Nonlinearity, 8 (1995), 1077.  doi: 10.1088/0951-7715/8/6/011.  Google Scholar [9] J. Cheeger and D. G. Ebin, "Comparison Theorems in Riemannian Geometry,", North-Holland Mathematical Library, 9 (1975).   Google Scholar [10] G. Contreras, Action potential and weak KAM solutions,, Calc. Var. Partial Differential Equations, 13 (2001), 427.   Google Scholar [11] G. Contreras and R. Ruggiero, Non-hyperbolic surfaces having all ideal triangles of finite area,, Bul. Braz. Math. Soc., 28 (1997), 43.  doi: 10.1007/BF01235988.  Google Scholar [12] G. Contreras and R. Iturriaga, Global minimizers of autonomous lagrangians,, to appear, (2004).   Google Scholar [13] G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values,, Geom. Funct. Anal., 8 (1998), 788.  doi: 10.1007/s000390050074.  Google Scholar [14] A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications,", Springer Verlag, (1998).   Google Scholar [15] P. Eberlein, Manifolds of nonpositive curvature,, in, 27 (1989), 223.   Google Scholar [16] R. Durrett, "Stochastic Calculus: A practical Introduction,", CRC-Press, (1996).   Google Scholar [17] L. C. Evans, Towards a quantum analog of weak KAM theory,, Commun. in Math. Phys., 244 (2004), 311.  doi: 10.1007/s00220-003-0975-5.  Google Scholar [18] A. Fathi, Weak KAM theorem and lagrangian dynamics,, to appear, (2004).   Google Scholar [19] A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls,, C. R. Acad. Sci. Paris Ser. I, 235 (1997), 649.   Google Scholar [20] A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation,, Invent. Math., 155 (2004), 363.  doi: 10.1007/s00222-003-0323-6.  Google Scholar [21] M. Gerber and V. Nitica, Holder exponents of horocycle foliations on surfaces,, Ergodic Theory Dynam. Systems, 19 (1999), 1247.  doi: 10.1017/S0143385799146832.  Google Scholar [22] D. A. Gomes and E. Valdinoci, Entropy penalization methods for Hamilton-Jacobi equations,, Adv. Math., 215 (2007), 94.  doi: 10.1016/j.aim.2007.04.001.  Google Scholar [23] D. A. Gomes, A. O. Lopes and J. Mohr, The Mather measure and a large deviation principle for the entropy penalized method,, to appear in Communications in Contemporary Mathematics., ().   Google Scholar [24] D. A. Gomes, A. O. Lopes and J. Mohr, Wigner measures and the semi-classical limit to the Aubry-Mather measure,, to appear, (2009).   Google Scholar [25] A. O. Lopes, V. Rosas and R. Ruggiero, Cohomology and Subcohomology for expansive geodesic flows,, Discrete and Continous Dynamical Systems, 17 (2007), 403.   Google Scholar [26] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems,, Nonlinearity, 9 (1996), 273.  doi: 10.1088/0951-7715/9/2/002.  Google Scholar [27] R. Markarian, Billiards with polynomial decay of correlations,, Ergodic Theory Dynam. Systems, 24 (2004), 177.  doi: 10.1017/S0143385703000270.  Google Scholar [28] D. Massart, Aubry sets vs Mather sets in two degrees of freedom,, to appear, (2008).   Google Scholar [29] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems,, Math. Z., 2 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar [30] S. Nonnenmacher, Some open questions in wave chaos,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/8/T01.  Google Scholar [31] R. Ruggiero., Expansive geodesic flows in manifolds with no conjugate points,, Ergodic Theory Dynam. Systems, 17 (1997), 211.  doi: 10.1017/S0143385797060963.  Google Scholar [32] R. Ruggiero, Expansive dynamics and hyperbolic geometry,, Bul. Braz. Math. Soc., 25 (1994), 139.  doi: 10.1007/BF01321305.  Google Scholar [33] O. M. Sarig., Subexponential decay of correlations,, Invent. Math., 150 (2002), 629.  doi: 10.1007/s00222-002-0248-5.  Google Scholar [34] D. Stroock, "Partial Differential Equations for Probabilists,", Cambridge Press, (2008).  doi: 10.1017/CBO9780511755255.  Google Scholar [35] L. S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153.  doi: 10.1007/BF02808180.  Google Scholar
 [1] Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379 [2] Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135 [3] Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807 [4] Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683 [5] Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823 [6] Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103 [7] Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983 [8] Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158 [9] Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245 [10] Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [11] Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61 [12] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [13] Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic & Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001 [14] Masayuki Asaoka, Kenichiro Yamamoto. On the large deviation rates of non-entropy-approachable measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4401-4410. doi: 10.3934/dcds.2013.33.4401 [15] Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048 [16] Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581 [17] Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867 [18] Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 [19] Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial & Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461 [20] Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

2018 Impact Factor: 1.143