单调回复关系中的脱钉力和Denjoy极小集
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摘要
我们考虑由r+1元生成函数S决定的单调回复关系。当r=1时,存在一个以S为生成函数的单调扭转映射。叶状结构(Foliation)对应于这个单调扭转映射的不变曲线。
在本文中,我们将给出判断叶状结构是否存在的标志量：正向脱钉力和负向脱钉力。具有无理旋转数的叶状结构存在当且仅当相应的正向或者负向脱钉力消失。对于有理数ω=q/p,正向或者负向脱钉力消失当且仅当在(p,q)周期的Birkhoff构型集中存在叶状结构。
如果某个单调扭转映射存在无理旋转数ω的不变曲线,则以ω为旋转数的Aubry-Mather集都在这条不变曲线上。我们证明在单调回复关系中也有类似的结果：如果存在某个无理旋转数的叶状结构,则在它之外没有同一旋转数的Denjoy极小集。实际上,无理旋转数的叶状结构上,Denjoy极小集最多只有一个。所以,不存在以ω为旋转数的叶状结构是存在多个ω旋转数的Denjoy极小集的必要条件。我们还将证明这个条件也是充分的。实际上,单调回复关系中无理旋转数的叶状结构是由Birkhoff的最小能量构型构成的。因此,上面的结论可以总结为：对无理数ω,存在多个ω旋转数的Denjoy极小集的充分必要条件是以ω为旋转数的最小能量构型不形成叶状结构。
We investigate the monotone recurrence relation generated by a function S of r+1variables, which is called generating function. For the monotone recurrence relation, itis significant to know the existence or non-existence of foliations for they correspondto the invariant circles of a twist map with generating function S in case r=1.
To give a criterion of existence or non-existence of foliations, we give two quantities:positive and negative depinning forces for each rotation number. We prove that thereis a foliation with irrational rotation number if and only if the corresponding positiveor negative depinning force vanishes. For a rational number ω=q/p, the positive ornegative depinning force vanishes if and only if there is a foliation contained in the(p, q)-periodic Birkhof configuration set.
It is well known that all Aubry-Mather sets of a twist map with irrational number ωlie on an invariant circle with rotation number ω. We prove that the similar conclusionholds for the monotone recurrence relation: There is no Denjoy minimal set withirrational rotation number outside the foliation with the same rotation number. Weprove that the number of Denjoy minimal sets contained in foliation is no more thanone. Hence, a necessary condition for the existence of distinct Denjoy minimal sets ofirrational rotation number is the foliation of the same rotation number is disintegrating.We shall show the condition is also a sufcient one. The foliation with irrationalrotation number for the monotone recurrence relation consists of Birkhof minimizers.Hence, the conclusion can be summarized as: There are many Denjoy minimal sets ofirrational rotation number ω if and only if the Birkhof minimizers of rotation numberω can not form a foliation.
在本文中,我们将给出判断叶状结构是否存在的标志量：正向脱钉力和负向脱钉力。具有无理旋转数的叶状结构存在当且仅当相应的正向或者负向脱钉力消失。对于有理数ω=q/p,正向或者负向脱钉力消失当且仅当在(p,q)周期的Birkhoff构型集中存在叶状结构。
如果某个单调扭转映射存在无理旋转数ω的不变曲线,则以ω为旋转数的Aubry-Mather集都在这条不变曲线上。我们证明在单调回复关系中也有类似的结果：如果存在某个无理旋转数的叶状结构,则在它之外没有同一旋转数的Denjoy极小集。实际上,无理旋转数的叶状结构上,Denjoy极小集最多只有一个。所以,不存在以ω为旋转数的叶状结构是存在多个ω旋转数的Denjoy极小集的必要条件。我们还将证明这个条件也是充分的。实际上,单调回复关系中无理旋转数的叶状结构是由Birkhoff的最小能量构型构成的。因此,上面的结论可以总结为：对无理数ω,存在多个ω旋转数的Denjoy极小集的充分必要条件是以ω为旋转数的最小能量构型不形成叶状结构。
We investigate the monotone recurrence relation generated by a function S of r+1variables, which is called generating function. For the monotone recurrence relation, itis significant to know the existence or non-existence of foliations for they correspondto the invariant circles of a twist map with generating function S in case r=1.
To give a criterion of existence or non-existence of foliations, we give two quantities:positive and negative depinning forces for each rotation number. We prove that thereis a foliation with irrational rotation number if and only if the corresponding positiveor negative depinning force vanishes. For a rational number ω=q/p, the positive ornegative depinning force vanishes if and only if there is a foliation contained in the(p, q)-periodic Birkhof configuration set.
It is well known that all Aubry-Mather sets of a twist map with irrational number ωlie on an invariant circle with rotation number ω. We prove that the similar conclusionholds for the monotone recurrence relation: There is no Denjoy minimal set withirrational rotation number outside the foliation with the same rotation number. Weprove that the number of Denjoy minimal sets contained in foliation is no more thanone. Hence, a necessary condition for the existence of distinct Denjoy minimal sets ofirrational rotation number is the foliation of the same rotation number is disintegrating.We shall show the condition is also a sufcient one. The foliation with irrationalrotation number for the monotone recurrence relation consists of Birkhof minimizers.Hence, the conclusion can be summarized as: There are many Denjoy minimal sets ofirrational rotation number ω if and only if the Birkhof minimizers of rotation numberω can not form a foliation.
引文
[1] S. Angenent, The periodic points of an area-preserving twist map, Comm. Math. Phys.,Vol.115(1988),353-374.
[2] S. Angenent, Monotone recurrence relations, their Birkhof orbits and topological en-tropy, Ergodic Theory Dynam. Systems, Vol.10(1990),15-41.
[3] V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Springer-Verlag,1989.
[4] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions,Phys. D, Vol.8(1983),381-422.
[5] M.Arnaud, The link between the shape of the irrational Aubry-Mather sets and theirLyapunov exponents, Ann. of Math., Vol.174(2011),1571-1601.
[6] C. Baesens and R. S. MacKay, Gradient dynamics of tilted Frenkel-Kontorova models,Nonlinearity, Vol.11(1998),949-964.
[7] V. Bangert, A uniqueness theorem forZn-periodic variational problems, Comment.Math. Helv., Vol.62(1987),511-531.
[8] V. Bangert, Mather sets for twist maps and geodesics on tori, in “Dynamics Reported”,Vol.1(1988),1-56, U. Kirchgraber and H. O. Walther eds., New York: Wiley.
[9] V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincar′e Anal. NonLin′eaire, Vol.6(1989),95-138.
[10] U. Bessi, Many solutions of elliptic problems onRnof irrational slope, Comm. PartialDiferential Equations, Vol.30(2005),1773-1804.
[11] G. D. Birkhof, Surface transformations and their dynamical applications. Acta Math.,Vol.43(1922),1-119.
[12] M. Blank, Metric properties of minimal solutions of discrete periodical variational prob-lems, Nonlinarity, Vol.2(1989),1-22.
[13] P. L. Boyland and G. R. Hall, Invariant and the ordered structure of periodic of orbitsin monotone twist map, Topology, Vol.6(1987),21-35.
[14] P. L. Boyland, Invariant circles and rotation bands in monotone twist maps, Comm.Math. Phys., Vol.113(1987),67-77.
[15] O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods, andApplications, Springer-Verlag,2004.
[16] W. Cheng and C. Q. Cheng, Existence of infinite non-Birkhof periodic orbits for area-preserving monotone twist maps of cylinders, Science in China, Vol.43(2000),810-817.
[17] S. N. Coppersmith and D. S. Fisher, Pinning transition of the discrete sine-Gordonequation, Phys. Rev. B, Vol.28(1983),2566-2581.
[18] S. N. Coppersmith and D. S. Fisher, Threshold behavior of a driven incommensurateharmonic chain, Phys. Rev. A, Vol.38(1988),6338-6350.
[19] R. de la Llave and E. Valdinoci, Critical points inside the gaps of ground state lamina-tions for some models in statistical mechanics, J. Stat. Phys., Vol.129(2007),81-119.
[20] R. de la Llave and E. Valdinoci, Ground states and critical points for generalized Frenkel-Kontorova models inZd, Nonlinearity, Vol.20(2007),2409-2424.
[21] R. de la Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media, Adv. Math., Vol.215(2007),379-426.
[22] R. de la Llave and E. Valdinoci, Ground states and critical points for Aubry-Mathertheory in statistical mechanics, J. Nonlinear Sci., Vol.20(2010),153-218.
[23] L. M. Flor′a and J. J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model, Adv.Phys., Vol.45(1996),505-598.
[24] E. Garibaldi and P. Thieullen, Minimizing orbits in the discrete Aubry Mather model,Nonlinearity, Vol.24(2011),563611.
[25] C. Gol′e, Ghost circles for twist maps, J. Difer. Equa., Vol.97(1992),140-173.
[26] C. Gol′e, A new proof of the Aubry-Mather’s theorem, Math. Z., Vol.210(1992),441-448.
[27] C. Gol′e, Symplectic Twist Maps: Global Variational Techniques, World Scientific, Sin-gapore,2001.
[28] D. L. Gorof, Hyperbolic sets for twist maps, Ergodic Theory Dynam. Systems,Vol.5(1985),337-339.
[29] I. Jungreis, A method for proving that monotone twist maps have no invariant circles,Ergodic Theory Dynam. Systems, Vol.11(1991),79-84.
[30] A. Katok, Some remarks on the Birkhof and Mather twist theorems, Ergodic TheoryDynam. Systems, Vol.2(1982),183-194.
[31] A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems,Cambridge University Press,1995.
[32] V. Knibbeler and B. Mramor and B. Rink, The laminations of a crystal near an anti-continuum limit, arrXiv:1305.7193v1.
[33] H. Koch, R. de la Llave, and C. Radin, Aubry-Mather theory for functions on lattices,Discrete Contin. Dyn. Syst., Vol.3(1997),135-151.
[34] A. N. Kolmogolov and S. V. Fomin, Elements of the Theory of Functions and FunctionalAnalysis, Dover Publications,1999.
[35] S. Lang, Real and Functional Analysis, Springer-Verlag, Third Edition,1993.
[36] R. S. MacKay and I.C. Percival, Converse KAM: theory and practice, Commun.Math.Phys., Vol.98(1985),496-512.
[37] R. S. MacKay, Scaling exponents at the transition by breaking of analyticity for incom-mensurate structures, Physica D, Vol.50(1991),71-79.
[38] R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for areapreserving maps, Nonlinearity, Vol.5(1992)867-888.
[39] J. N. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annu-lus, Topology, Vol.21(1982),457-467.
[40] J. N. Mather, Non-existence of invariant circles, Ergodic Theory Dynam. Systems,Vol.4(1984),301-309.
[41] J. N. Mather, More Denjoy minimal sets for area preserving difeomorphisms, Comment.Math. Helv., Vol.60(1985),508-557.
[42] J. N. Mather, A criterion for the non-existence of invariant circles, Publ. Math. IHES,Vol.63(1986),153-204.
[43] J. N. Mather and F. Forni, Action minimizing orbits in hamiltomian systems, Tran-sition to Chaos in Classical and Quantum Mechanics Lecture Notes in Mathematics,Vol.1589(1994),92-186.
[44] J. D. Messi, Symplectic maps, variational principles, and transport, Review on modernphiysics, Vol.64(1992),795-848.
[45] J. Moser, Stable and Random Motion in Dynamical Systems, Ann. of Math. Studies77,Princeton Univ. Press,1973.
[46] J. Moser, Recent developments in the theory of Hamiltonian systems, SIAM Review,Vol.28(1986),459-485.
[47] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincar′eAnal. Non Lin′eaire, Vol.3(1986),229-272.
[48] B. Mramor and B. Rink, Ghost circles in lattice Aubry-Mather theory, J. DiferentialEquations, Vol.252(2012),3163-3208.
[49] B. Mramor and B. Rink, A dichotomy theorem for minimizers of monotone recurrencerelations, acccepted by Ergodic Theory Dynam. Systems, arrXiv:1111.5970v2.
[50] B. Mramor and B. Rink, On the destruction of minimal foliations, arrXiv:1111.5966v1.
[51] B. Mramor and B. Rink, Continuity of the Peierls barrier and robustness of laminations,arrXiv:1308.3073v1.
[52] B. Mramor, PhD thesis: Monotone variational recurrence relations,2012.
[53] J. R. Munkres, Topology, Second Edition, Pearson Prentice Hall,1975.
[54]W.-X. Qin, Dynamics of the Frenkel-Kontorova model with irrational mean spacing, Nonlinearity, Vol.23(2010),1873-1886.
[55]W. M. Senn, Phase-locking in the multidimensional Frenkel-Kontorova model, Math. Z., Vol.227(1998),623"643.
[56]S. Slijepcevic, The pulled Frenkel-Kontorova chain, Nonlinearity, Vol.11(1998),923-948.
[57]J. Stark, An exhausitive criterion for the non-existence of invariant circles for area-preserving twist maps, Commun. Math. Phys., Vol.117(1988),177-189.
[58]蒋美跃,裴明亮,扭转映射的Aubry-Mather理论及其应用,数学进展,vol.23(1994),96-112.
[59]张恭庆,泛函分析讲义,北京大学出版社,1987.
[60]张筑生,微分动力系统原理,科学出版社,1987.
[61]钟承奎,非线性泛函分析引论,兰州大学出版社,1998.
[2] S. Angenent, Monotone recurrence relations, their Birkhof orbits and topological en-tropy, Ergodic Theory Dynam. Systems, Vol.10(1990),15-41.
[3] V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Springer-Verlag,1989.
[4] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions,Phys. D, Vol.8(1983),381-422.
[5] M.Arnaud, The link between the shape of the irrational Aubry-Mather sets and theirLyapunov exponents, Ann. of Math., Vol.174(2011),1571-1601.
[6] C. Baesens and R. S. MacKay, Gradient dynamics of tilted Frenkel-Kontorova models,Nonlinearity, Vol.11(1998),949-964.
[7] V. Bangert, A uniqueness theorem forZn-periodic variational problems, Comment.Math. Helv., Vol.62(1987),511-531.
[8] V. Bangert, Mather sets for twist maps and geodesics on tori, in “Dynamics Reported”,Vol.1(1988),1-56, U. Kirchgraber and H. O. Walther eds., New York: Wiley.
[9] V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincar′e Anal. NonLin′eaire, Vol.6(1989),95-138.
[10] U. Bessi, Many solutions of elliptic problems onRnof irrational slope, Comm. PartialDiferential Equations, Vol.30(2005),1773-1804.
[11] G. D. Birkhof, Surface transformations and their dynamical applications. Acta Math.,Vol.43(1922),1-119.
[12] M. Blank, Metric properties of minimal solutions of discrete periodical variational prob-lems, Nonlinarity, Vol.2(1989),1-22.
[13] P. L. Boyland and G. R. Hall, Invariant and the ordered structure of periodic of orbitsin monotone twist map, Topology, Vol.6(1987),21-35.
[14] P. L. Boyland, Invariant circles and rotation bands in monotone twist maps, Comm.Math. Phys., Vol.113(1987),67-77.
[15] O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model, Concepts, Methods, andApplications, Springer-Verlag,2004.
[16] W. Cheng and C. Q. Cheng, Existence of infinite non-Birkhof periodic orbits for area-preserving monotone twist maps of cylinders, Science in China, Vol.43(2000),810-817.
[17] S. N. Coppersmith and D. S. Fisher, Pinning transition of the discrete sine-Gordonequation, Phys. Rev. B, Vol.28(1983),2566-2581.
[18] S. N. Coppersmith and D. S. Fisher, Threshold behavior of a driven incommensurateharmonic chain, Phys. Rev. A, Vol.38(1988),6338-6350.
[19] R. de la Llave and E. Valdinoci, Critical points inside the gaps of ground state lamina-tions for some models in statistical mechanics, J. Stat. Phys., Vol.129(2007),81-119.
[20] R. de la Llave and E. Valdinoci, Ground states and critical points for generalized Frenkel-Kontorova models inZd, Nonlinearity, Vol.20(2007),2409-2424.
[21] R. de la Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media, Adv. Math., Vol.215(2007),379-426.
[22] R. de la Llave and E. Valdinoci, Ground states and critical points for Aubry-Mathertheory in statistical mechanics, J. Nonlinear Sci., Vol.20(2010),153-218.
[23] L. M. Flor′a and J. J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model, Adv.Phys., Vol.45(1996),505-598.
[24] E. Garibaldi and P. Thieullen, Minimizing orbits in the discrete Aubry Mather model,Nonlinearity, Vol.24(2011),563611.
[25] C. Gol′e, Ghost circles for twist maps, J. Difer. Equa., Vol.97(1992),140-173.
[26] C. Gol′e, A new proof of the Aubry-Mather’s theorem, Math. Z., Vol.210(1992),441-448.
[27] C. Gol′e, Symplectic Twist Maps: Global Variational Techniques, World Scientific, Sin-gapore,2001.
[28] D. L. Gorof, Hyperbolic sets for twist maps, Ergodic Theory Dynam. Systems,Vol.5(1985),337-339.
[29] I. Jungreis, A method for proving that monotone twist maps have no invariant circles,Ergodic Theory Dynam. Systems, Vol.11(1991),79-84.
[30] A. Katok, Some remarks on the Birkhof and Mather twist theorems, Ergodic TheoryDynam. Systems, Vol.2(1982),183-194.
[31] A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems,Cambridge University Press,1995.
[32] V. Knibbeler and B. Mramor and B. Rink, The laminations of a crystal near an anti-continuum limit, arrXiv:1305.7193v1.
[33] H. Koch, R. de la Llave, and C. Radin, Aubry-Mather theory for functions on lattices,Discrete Contin. Dyn. Syst., Vol.3(1997),135-151.
[34] A. N. Kolmogolov and S. V. Fomin, Elements of the Theory of Functions and FunctionalAnalysis, Dover Publications,1999.
[35] S. Lang, Real and Functional Analysis, Springer-Verlag, Third Edition,1993.
[36] R. S. MacKay and I.C. Percival, Converse KAM: theory and practice, Commun.Math.Phys., Vol.98(1985),496-512.
[37] R. S. MacKay, Scaling exponents at the transition by breaking of analyticity for incom-mensurate structures, Physica D, Vol.50(1991),71-79.
[38] R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for areapreserving maps, Nonlinearity, Vol.5(1992)867-888.
[39] J. N. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annu-lus, Topology, Vol.21(1982),457-467.
[40] J. N. Mather, Non-existence of invariant circles, Ergodic Theory Dynam. Systems,Vol.4(1984),301-309.
[41] J. N. Mather, More Denjoy minimal sets for area preserving difeomorphisms, Comment.Math. Helv., Vol.60(1985),508-557.
[42] J. N. Mather, A criterion for the non-existence of invariant circles, Publ. Math. IHES,Vol.63(1986),153-204.
[43] J. N. Mather and F. Forni, Action minimizing orbits in hamiltomian systems, Tran-sition to Chaos in Classical and Quantum Mechanics Lecture Notes in Mathematics,Vol.1589(1994),92-186.
[44] J. D. Messi, Symplectic maps, variational principles, and transport, Review on modernphiysics, Vol.64(1992),795-848.
[45] J. Moser, Stable and Random Motion in Dynamical Systems, Ann. of Math. Studies77,Princeton Univ. Press,1973.
[46] J. Moser, Recent developments in the theory of Hamiltonian systems, SIAM Review,Vol.28(1986),459-485.
[47] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincar′eAnal. Non Lin′eaire, Vol.3(1986),229-272.
[48] B. Mramor and B. Rink, Ghost circles in lattice Aubry-Mather theory, J. DiferentialEquations, Vol.252(2012),3163-3208.
[49] B. Mramor and B. Rink, A dichotomy theorem for minimizers of monotone recurrencerelations, acccepted by Ergodic Theory Dynam. Systems, arrXiv:1111.5970v2.
[50] B. Mramor and B. Rink, On the destruction of minimal foliations, arrXiv:1111.5966v1.
[51] B. Mramor and B. Rink, Continuity of the Peierls barrier and robustness of laminations,arrXiv:1308.3073v1.
[52] B. Mramor, PhD thesis: Monotone variational recurrence relations,2012.
[53] J. R. Munkres, Topology, Second Edition, Pearson Prentice Hall,1975.
[54]W.-X. Qin, Dynamics of the Frenkel-Kontorova model with irrational mean spacing, Nonlinearity, Vol.23(2010),1873-1886.
[55]W. M. Senn, Phase-locking in the multidimensional Frenkel-Kontorova model, Math. Z., Vol.227(1998),623"643.
[56]S. Slijepcevic, The pulled Frenkel-Kontorova chain, Nonlinearity, Vol.11(1998),923-948.
[57]J. Stark, An exhausitive criterion for the non-existence of invariant circles for area-preserving twist maps, Commun. Math. Phys., Vol.117(1988),177-189.
[58]蒋美跃,裴明亮,扭转映射的Aubry-Mather理论及其应用,数学进展,vol.23(1994),96-112.
[59]张恭庆,泛函分析讲义,北京大学出版社,1987.
[60]张筑生,微分动力系统原理,科学出版社,1987.
[61]钟承奎,非线性泛函分析引论,兰州大学出版社,1998.