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On the Cauchy problem for nonlinear Schrödinger equations with rotation
Localized asymptotic behavior for almost additive potentials
1. | LAGA (UMR 7539), Département de Mathématiques, Institut Galilée, Université Paris 13, Villetaneuse, France |
2. | Department of Mathematics, Tsinghua University, Beijing, China |
References:
[1] |
J. Barral and D. J. Feng, Weighted thermodynamic formalism and applications, arXiv:0909.4247v1. |
[2] |
L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Sys., 16 (1996), 871-927.
doi: 10.1017/S0143385700010117. |
[3] |
L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Contin. Dyn. Syst., 16 (2006), 279-305.
doi: 10.3934/dcds.2006.16.279. |
[4] |
L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl., 92 (2009), 1-17.
doi: 10.1016/j.matpur.2009.04.006. |
[5] |
L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371.
doi: 10.1007/s002200000268. |
[6] |
L. Barreira, B. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, J. Math. Pures Appl. (9), 81 (2002), 67-91.
doi: 10.1016/S0021-7824(01)01228-4. |
[7] |
R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. |
[8] |
R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. |
[9] |
R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340.
doi: 10.1215/S0012-7094-40-00718-9. |
[10] |
G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures, J. Stat. Phys., 66 (1992), 775-790.
doi: 10.1007/BF01055700. |
[11] |
Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. |
[12] |
P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys., 47 (1987), 609-644.
doi: 10.1007/BF01206149. |
[13] |
K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers, J. Phys. A, 21 (1988), L737-L742.
doi: 10.1088/0305-4470/21/14/005. |
[14] |
A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space, J. Statist. Phys., 99 (2000), 813-856.
doi: 10.1023/A:1018643512559. |
[15] |
A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.
doi: 10.1017/S0024610701002137. |
[16] |
D.-J. Feng, Lyapounov exponents for products of matrices and multifractal analysis. Part I: Positive matrices, Israel J. Math., 138 (2003), 353-376.
doi: 10.1007/BF02783432. |
[17] |
D.-J. Feng, The variational principle for products of non-negative matrices, Nonlinearity, 17 (2004), 447-457.
doi: 10.1088/0951-7715/17/2/004. |
[18] |
D.-J. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297 (2010), 1-43.
doi: 10.1007/s00220-010-1031-x. |
[19] |
D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378. |
[20] |
D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.
doi: 10.1006/aima.2001.2054. |
[21] |
D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. Dynam. Sys., 17 (1997), 147-167.
doi: 10.1017/S0143385797060987. |
[22] |
M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra, Nonlinearity, 14 (2001), 395-409.
doi: 10.1088/0951-7715/14/2/312. |
[23] |
M. Kesseböhmer and B. Stratmann, A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergod. Th. Dynam. Sys., 24 (2004), 141-170. |
[24] |
N. G. Makarov, Fine structure of harmonic measure, St. Peterburg Math. J., 10 (1999), 217-268. |
[25] |
A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454.
doi: 10.3934/dcds.2006.16.435. |
[26] |
E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures, Nonlinearity, 12 (1999), 1571-1585.
doi: 10.1088/0951-7715/12/6/309. |
[27] |
L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.
doi: 10.1016/j.matpur.2003.09.007. |
[28] |
Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc., 129 (2001), 2689-2699.
doi: 10.1090/S0002-9939-01-05969-X. |
[29] |
Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. |
[30] |
Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys., 86 (1997), 233-275.
doi: 10.1007/BF02180206. |
[31] |
Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical Foundation, and Examples, Chaos, 7 (1997), 89-106.
doi: 10.1063/1.166242. |
[32] |
D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Th. Dynam. Sys., 9 (1989), 527-541. |
[33] |
D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. |
show all references
References:
[1] |
J. Barral and D. J. Feng, Weighted thermodynamic formalism and applications, arXiv:0909.4247v1. |
[2] |
L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Sys., 16 (1996), 871-927.
doi: 10.1017/S0143385700010117. |
[3] |
L. Barreira, Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures, Discrete Contin. Dyn. Syst., 16 (2006), 279-305.
doi: 10.3934/dcds.2006.16.279. |
[4] |
L. Barreira and P. Doutor, Almost additive multifractal analysis, J. Math. Pures Appl., 92 (2009), 1-17.
doi: 10.1016/j.matpur.2009.04.006. |
[5] |
L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys., 214 (2000), 339-371.
doi: 10.1007/s002200000268. |
[6] |
L. Barreira, B. Saussol and J. Schmeling, Higher-dimensional multifractal analysis, J. Math. Pures Appl. (9), 81 (2002), 67-91.
doi: 10.1016/S0021-7824(01)01228-4. |
[7] |
R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. |
[8] |
R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. |
[9] |
R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340.
doi: 10.1215/S0012-7094-40-00718-9. |
[10] |
G. Brown, G. Michon and J. Peyrière, On the multifractal analysis of measures, J. Stat. Phys., 66 (1992), 775-790.
doi: 10.1007/BF01055700. |
[11] |
Y.-L. Cao, D.-J. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657. |
[12] |
P. Collet, J. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys., 47 (1987), 609-644.
doi: 10.1007/BF01206149. |
[13] |
K. J. Falconer, A subadditive thermodynamic formalism for mixing repellers, J. Phys. A, 21 (1988), L737-L742.
doi: 10.1088/0305-4470/21/14/005. |
[14] |
A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space, J. Statist. Phys., 99 (2000), 813-856.
doi: 10.1023/A:1018643512559. |
[15] |
A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.
doi: 10.1017/S0024610701002137. |
[16] |
D.-J. Feng, Lyapounov exponents for products of matrices and multifractal analysis. Part I: Positive matrices, Israel J. Math., 138 (2003), 353-376.
doi: 10.1007/BF02783432. |
[17] |
D.-J. Feng, The variational principle for products of non-negative matrices, Nonlinearity, 17 (2004), 447-457.
doi: 10.1088/0951-7715/17/2/004. |
[18] |
D.-J. Feng and W. Huang, Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297 (2010), 1-43.
doi: 10.1007/s00220-010-1031-x. |
[19] |
D.-J. Feng and K.-S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002), 363-378. |
[20] |
D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.
doi: 10.1006/aima.2001.2054. |
[21] |
D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. Dynam. Sys., 17 (1997), 147-167.
doi: 10.1017/S0143385797060987. |
[22] |
M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra, Nonlinearity, 14 (2001), 395-409.
doi: 10.1088/0951-7715/14/2/312. |
[23] |
M. Kesseböhmer and B. Stratmann, A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergod. Th. Dynam. Sys., 24 (2004), 141-170. |
[24] |
N. G. Makarov, Fine structure of harmonic measure, St. Peterburg Math. J., 10 (1999), 217-268. |
[25] |
A. Mummert, The thermodynamic formalism for almost-additive sequences, Discrete Contin. Dyn. Syst., 16 (2006), 435-454.
doi: 10.3934/dcds.2006.16.435. |
[26] |
E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures, Nonlinearity, 12 (1999), 1571-1585.
doi: 10.1088/0951-7715/12/6/309. |
[27] |
L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.
doi: 10.1016/j.matpur.2003.09.007. |
[28] |
Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc., 129 (2001), 2689-2699.
doi: 10.1090/S0002-9939-01-05969-X. |
[29] |
Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. |
[30] |
Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys., 86 (1997), 233-275.
doi: 10.1007/BF02180206. |
[31] |
Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, Mathematical Foundation, and Examples, Chaos, 7 (1997), 89-106.
doi: 10.1063/1.166242. |
[32] |
D. A. Rand, The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Th. Dynam. Sys., 9 (1989), 527-541. |
[33] |
D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. |
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