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

Artur O. Lopes; Rafael O. Ruggiero

doi:10.3934/dcds.2011.29.1155

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

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

Artur O. Lopes ^1, and Rafael O. Ruggiero ^2,

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

Abstract
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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.

Keywords: twisted Brownian motion., large deviation, Geodesic flow, Aubry-Mather measure.

Mathematics Subject Classification: 37D40, 37C50, 37A50, 37D05, 53D2.

Citation: Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems, 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), 207-276. doi: 10.4171/JEMS/9. Google Scholar
[2]	N. Anantharaman, Counting geodesics which are optimal in homology, Erg. Theo. and Dyn. Syst., 23 (2003), 353-388. doi: 10.1017/S0143385702001372. Google Scholar
[3]	N. Anantharaman, Entropie et localisation des fonctions propres, ENS-Lyon, (2006), http://www.math.polytechnique.fr/ nalini/HDR.pdf. 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), 513-528. 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), 503-511. 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), 1077-1085 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, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American-Elsevier Publishing Co., Inc., New York, 1975. Google Scholar
[10]	G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. Google Scholar
[11]	G. Contreras and R. Ruggiero, Non-hyperbolic surfaces having all ideal triangles of finite area, Bul. Braz. Math. Soc., 28 (1997), 43-71. 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), 788-809. 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 "Global Differential Geometry," MAA Stud. Math., 27, Math. Assoc. America, Washington DC, (1989), 223-258. 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), 311-334. 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), 649-652. Google Scholar
[20]	A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. 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), 1247-1254. doi: 10.1017/S0143385799146832. Google Scholar
[22]	D. A. Gomes and E. Valdinoci, Entropy penalization methods for Hamilton-Jacobi equations, Adv. Math., 215 (2007), 94-152. 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), 403-422. Google Scholar
[26]	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. Google Scholar
[27]	R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems, 24 (2004), 177-197 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), 169-207. doi: 10.1007/BF02571383. Google Scholar
[30]	S. Nonnenmacher, Some open questions in wave chaos, Nonlinearity, 21 (2008), T113-T121. 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), 211-225. doi: 10.1017/S0143385797060963. Google Scholar
[32]	R. Ruggiero, Expansive dynamics and hyperbolic geometry, Bul. Braz. Math. Soc., 25 (1994), 139-172. doi: 10.1007/BF01321305. Google Scholar
[33]	O. M. Sarig., Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. 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), 153-188. 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), 207-276. doi: 10.4171/JEMS/9. Google Scholar
[2]	N. Anantharaman, Counting geodesics which are optimal in homology, Erg. Theo. and Dyn. Syst., 23 (2003), 353-388. doi: 10.1017/S0143385702001372. Google Scholar
[3]	N. Anantharaman, Entropie et localisation des fonctions propres, ENS-Lyon, (2006), http://www.math.polytechnique.fr/ nalini/HDR.pdf. 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), 513-528. 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), 503-511. 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), 1077-1085 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, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American-Elsevier Publishing Co., Inc., New York, 1975. Google Scholar
[10]	G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. Google Scholar
[11]	G. Contreras and R. Ruggiero, Non-hyperbolic surfaces having all ideal triangles of finite area, Bul. Braz. Math. Soc., 28 (1997), 43-71. 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), 788-809. 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 "Global Differential Geometry," MAA Stud. Math., 27, Math. Assoc. America, Washington DC, (1989), 223-258. 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), 311-334. 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), 649-652. Google Scholar
[20]	A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388. 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), 1247-1254. doi: 10.1017/S0143385799146832. Google Scholar
[22]	D. A. Gomes and E. Valdinoci, Entropy penalization methods for Hamilton-Jacobi equations, Adv. Math., 215 (2007), 94-152. 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), 403-422. Google Scholar
[26]	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. Google Scholar
[27]	R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems, 24 (2004), 177-197 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), 169-207. doi: 10.1007/BF02571383. Google Scholar
[30]	S. Nonnenmacher, Some open questions in wave chaos, Nonlinearity, 21 (2008), T113-T121. 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), 211-225. doi: 10.1017/S0143385797060963. Google Scholar
[32]	R. Ruggiero, Expansive dynamics and hyperbolic geometry, Bul. Braz. Math. Soc., 25 (1994), 139-172. doi: 10.1007/BF01321305. Google Scholar
[33]	O. M. Sarig., Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. 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), 153-188. doi: 10.1007/BF02808180. Google Scholar

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