-
Previous Article
Computational hyperbolicity
- DCDS Home
- This Issue
-
Next Article
Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups
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 |
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. |
[1] |
Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983 |
[8] |
Kaizhi Wang, Lin Wang, Jun Yan. Aubry-Mather theory for contact Hamiltonian systems II. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 555-595. doi: 10.3934/dcds.2021128 |
[9] |
Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158 |
[10] |
Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245 |
[11] |
Francesca Biagini, Thilo Meyer-Brandis, Bernt Øksendal, Krzysztof Paczka. Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 8-. doi: 10.1186/s41546-018-0033-z |
[12] |
Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 |
[13] |
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 |
[14] |
Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic and Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001 |
[15] |
Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 |
[16] |
Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867 |
[17] |
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 |
[18] |
Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]