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

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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 and 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.
[2]	N. Anantharaman, Counting geodesics which are optimal in homology, Erg. Theo. and Dyn. Syst., 23 (2003), 353-388. doi: 10.1017/S0143385702001372.
[3]	N. Anantharaman, Entropie et localisation des fonctions propres, ENS-Lyon, (2006), http://www.math.polytechnique.fr/ nalini/HDR.pdf.
[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.
[5]	W. Ballman, M. Gromov and V. Schroeder, "Manifolds of Non-positive Curvature," Birkhausser, 1985.
[6]	P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.
[7]	H. Busemann, "The Geometry of Geodesics," Academic Press, 1955.
[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.
[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.
[10]	G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458.
[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.
[12]	G. Contreras and R. Iturriaga, Global minimizers of autonomous lagrangians, to appear, (2004).
[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.
[14]	A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Springer Verlag, 1998.
[15]	P. Eberlein, Manifolds of nonpositive curvature, in "Global Differential Geometry," MAA Stud. Math., 27, Math. Assoc. America, Washington DC, (1989), 223-258.
[16]	R. Durrett, "Stochastic Calculus: A practical Introduction," CRC-Press, 1996.
[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.
[18]	A. Fathi, Weak KAM theorem and lagrangian dynamics, to appear, 2004.
[19]	A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls, C. R. Acad. Sci. Paris Ser. I, 235 (1997), 649-652.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems, 24 (2004), 177-197 doi: 10.1017/S0143385703000270.
[28]	D. Massart, Aubry sets vs Mather sets in two degrees of freedom, to appear, 2008.
[29]	J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 2 (1991), 169-207. doi: 10.1007/BF02571383.
[30]	S. Nonnenmacher, Some open questions in wave chaos, Nonlinearity, 21 (2008), T113-T121. doi: 10.1088/0951-7715/21/8/T01.
[31]	R. Ruggiero., Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225. doi: 10.1017/S0143385797060963.
[32]	R. Ruggiero, Expansive dynamics and hyperbolic geometry, Bul. Braz. Math. Soc., 25 (1994), 139-172. doi: 10.1007/BF01321305.
[33]	O. M. Sarig., Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. doi: 10.1007/s00222-002-0248-5.
[34]	D. Stroock, "Partial Differential Equations for Probabilists," Cambridge Press, 2008. doi: 10.1017/CBO9780511755255.
[35]	L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.

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.
[2]	N. Anantharaman, Counting geodesics which are optimal in homology, Erg. Theo. and Dyn. Syst., 23 (2003), 353-388. doi: 10.1017/S0143385702001372.
[3]	N. Anantharaman, Entropie et localisation des fonctions propres, ENS-Lyon, (2006), http://www.math.polytechnique.fr/ nalini/HDR.pdf.
[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.
[5]	W. Ballman, M. Gromov and V. Schroeder, "Manifolds of Non-positive Curvature," Birkhausser, 1985.
[6]	P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.
[7]	H. Busemann, "The Geometry of Geodesics," Academic Press, 1955.
[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.
[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.
[10]	G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458.
[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.
[12]	G. Contreras and R. Iturriaga, Global minimizers of autonomous lagrangians, to appear, (2004).
[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.
[14]	A. Dembo and O. Zeitouni, "Large Deviations Techniques and Applications," Springer Verlag, 1998.
[15]	P. Eberlein, Manifolds of nonpositive curvature, in "Global Differential Geometry," MAA Stud. Math., 27, Math. Assoc. America, Washington DC, (1989), 223-258.
[16]	R. Durrett, "Stochastic Calculus: A practical Introduction," CRC-Press, 1996.
[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.
[18]	A. Fathi, Weak KAM theorem and lagrangian dynamics, to appear, 2004.
[19]	A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls, C. R. Acad. Sci. Paris Ser. I, 235 (1997), 649-652.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems, 24 (2004), 177-197 doi: 10.1017/S0143385703000270.
[28]	D. Massart, Aubry sets vs Mather sets in two degrees of freedom, to appear, 2008.
[29]	J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 2 (1991), 169-207. doi: 10.1007/BF02571383.
[30]	S. Nonnenmacher, Some open questions in wave chaos, Nonlinearity, 21 (2008), T113-T121. doi: 10.1088/0951-7715/21/8/T01.
[31]	R. Ruggiero., Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225. doi: 10.1017/S0143385797060963.
[32]	R. Ruggiero, Expansive dynamics and hyperbolic geometry, Bul. Braz. Math. Soc., 25 (1994), 139-172. doi: 10.1007/BF01321305.
[33]	O. M. Sarig., Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653. doi: 10.1007/s00222-002-0248-5.
[34]	D. Stroock, "Partial Differential Equations for Probabilists," Cambridge Press, 2008. doi: 10.1017/CBO9780511755255.
[35]	L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.

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