与复动力系统相关的(?)_b及Schroeder方程的研究
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摘要
本文我们主要讨论与复动力系统相关的两种方程的求解,一种是抛物黎曼曲面迭片结构(lamination)上(?)方程的求解,另一种是多圆柱上Schroeder方程的求解。其包含四部分内容：
首先在第二章介绍了与复几何有关的基本概念。我们主要讨论了复流形上的切空间及其对偶空间,全纯向量丛,Hermitian度量,Kobayashi度量,全纯向量丛上的联络、曲率以及与(?)相关的算子。
第三章讨论了黎曼曲面迭片结构的相关概念及其上的万有覆盖族。介绍了其上的CR线丛,以及切向(?)算子,采用抛物黎曼曲面迭片结构的仿射结构连续性,得到了其上万有覆盖族的一致正则性,也就是找到了连续且在传导方向上局部一致收敛的万有覆盖族。
第四章我们具体在抛物黎曼曲面迭片结构上求解(?)方程。首先给出了复平面上的一些相关估计,接着在平凡抛物黎曼曲面迭片结构C×T上用Hormander的方法求解了(?)方程,并得到了连续解。接着给出了主要定理4.1.1的证明。在本章的最后我们着重讨论了抛物黎曼曲面迭片结构的两个特例——环面纬垂(suspension)和环面射影极限——上(?)方程的求解。
第五章讨论了多圆柱上Schroeder方程的求解。首先给出了一个弱化Schroeder方程的求解,介绍了多圆柱上Julia-Wolff-Caratheodory定理。采用该定理我们找到了φ能够求解Schroeder方程的的一个充分条件。然后我们分情况讨论求解了Schroeder方程。
In this thesis, we mainly discuss the solutions of two kinds of equations associated with complex dynamic systems, one is the solution of (?)b equation on parabolic Riemann surface lam-inations, the other is the solution of Schroeder equation on polydisk. The content consists of the following four parts:
In Section2, we introduce some basic concepts which is associated with complex geome-try. We mainly discuss the concepts of tangent space and its dual space on complex manifold, holomorphic vector bundles, Hermitian metric, Kobayashi metric, connections and curvature on holomorphic vector bundles and the operators associated to (?).
In Section3, we discuss some related concepts of Riemann surface lamination and universal coverings on it. We introduce CR line bundles and tangent (?)b operator on it, using continuity of affine structure of parabolic Riemann surface lamination, we get transversal regularity of uni-versal coverings, that is we find universal coverings which is continuous and locally uniformly convergent in the transversal direction.
In Section4, we actually solve the (?)b equation on parabolic Riemann surface laminations. We first give some estimates on complex plane, then we solve (?)b equation by the Hormander's method on trival parabolic Riemann surface lamination C x T and get a continuous solution. We then give proof of the main Theorem4.1.1. At the end we give a discussion of solutions of equations on two special cases of parabolic Riemann surface laminations-suspensions of torus and projective limits of torus.
In Section5, we discuss the solution of Schroeder equation on polydisk. First we give a solution of weak Schroeder equation and Julia-Wolff-Caratheodory Theorem on polydisk. Using the theorem we find a sufficient condition of φ to make Schroeder equation solvable. And then we discuss the solution of Schroeder equation in several cases.
首先在第二章介绍了与复几何有关的基本概念。我们主要讨论了复流形上的切空间及其对偶空间,全纯向量丛,Hermitian度量,Kobayashi度量,全纯向量丛上的联络、曲率以及与(?)相关的算子。
第三章讨论了黎曼曲面迭片结构的相关概念及其上的万有覆盖族。介绍了其上的CR线丛,以及切向(?)算子,采用抛物黎曼曲面迭片结构的仿射结构连续性,得到了其上万有覆盖族的一致正则性,也就是找到了连续且在传导方向上局部一致收敛的万有覆盖族。
第四章我们具体在抛物黎曼曲面迭片结构上求解(?)方程。首先给出了复平面上的一些相关估计,接着在平凡抛物黎曼曲面迭片结构C×T上用Hormander的方法求解了(?)方程,并得到了连续解。接着给出了主要定理4.1.1的证明。在本章的最后我们着重讨论了抛物黎曼曲面迭片结构的两个特例——环面纬垂(suspension)和环面射影极限——上(?)方程的求解。
第五章讨论了多圆柱上Schroeder方程的求解。首先给出了一个弱化Schroeder方程的求解,介绍了多圆柱上Julia-Wolff-Caratheodory定理。采用该定理我们找到了φ能够求解Schroeder方程的的一个充分条件。然后我们分情况讨论求解了Schroeder方程。
In this thesis, we mainly discuss the solutions of two kinds of equations associated with complex dynamic systems, one is the solution of (?)b equation on parabolic Riemann surface lam-inations, the other is the solution of Schroeder equation on polydisk. The content consists of the following four parts:
In Section2, we introduce some basic concepts which is associated with complex geome-try. We mainly discuss the concepts of tangent space and its dual space on complex manifold, holomorphic vector bundles, Hermitian metric, Kobayashi metric, connections and curvature on holomorphic vector bundles and the operators associated to (?).
In Section3, we discuss some related concepts of Riemann surface lamination and universal coverings on it. We introduce CR line bundles and tangent (?)b operator on it, using continuity of affine structure of parabolic Riemann surface lamination, we get transversal regularity of uni-versal coverings, that is we find universal coverings which is continuous and locally uniformly convergent in the transversal direction.
In Section4, we actually solve the (?)b equation on parabolic Riemann surface laminations. We first give some estimates on complex plane, then we solve (?)b equation by the Hormander's method on trival parabolic Riemann surface lamination C x T and get a continuous solution. We then give proof of the main Theorem4.1.1. At the end we give a discussion of solutions of equations on two special cases of parabolic Riemann surface laminations-suspensions of torus and projective limits of torus.
In Section5, we discuss the solution of Schroeder equation on polydisk. First we give a solution of weak Schroeder equation and Julia-Wolff-Caratheodory Theorem on polydisk. Using the theorem we find a sufficient condition of φ to make Schroeder equation solvable. And then we discuss the solution of Schroeder equation in several cases.
引文
[1]J. E. Fornaess, N. Sibony, and E. F. Wold, Examples of minimal laminations and associated currents, Math. Zeit.269 (2011) 495-520
[2]T. Ohsawa, On projectively embeddable complex-foliated structure, Publ. Res. Inst. Math. Sci.48 (2012), no.3,735-747.
[3]J. E. Fornaess, E. F. Wold, Solving (?)b on hyperbolic laminations, preprint,2011. arX-iv:1108.2286.
[4]G. Valiron, Sur l'iteration des fonctions holomorphes dans un demi-plan, Bull. Sci. Math. 55(2) (1931),105-128.
[5]F. Bracci, G. Gentili, P. P. Corradini, Valiron's construction in higher dimension, Rev. Mat. Iber. (1) 26 (2010),57-76.
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[7]A. Candel, Uniformization of surface laminations, Ann.Sci.Ecole Norm.Sup.26(4) (1993), 489-516.
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[9]P. S. Bourdon, Density of the polynomials in Bergman spaces, Pacific J. Math.130 (1987), 215-221.
[10]C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Ame. Math. Soc.265 (1981),69-95.
[11]Ch. Pommerenke, On asymptotic iteration of anslytic functions in the disk, Analysis 1. (1981),45-61.
[12]M. Abate, The Julia-Wolff-Cratheodory theorem in poly disks, Jour, d'Anal. Math.74 (1998),275-306.
[13]M. Abate, Horospheres and iterates of holomorphic maps, Math. Z.198 (1988),225-238.
[14]M. Abate, Common fixed points of commuting holomorphic maps, Math. Ann.283 (1989), 645-655.
[15]M. Abate, Angular derivatives in strongly pseudoconvex domains, Proc. Symp. Pure Math. 52(2) (1991),23-40.
[16]W. Abikhow, Real Analytic Theory of Teichmuller Space, Lect. Notes in Math.,820, Springer-Verlag 1980.
[17]L. Ahlfors, Lectures on Quasiconformal Mappings. Wadsworth 1987.
[18]L. Ahlfors, L. Bers, Riemann's mapping theorem for variable metrics, Math. Ann.72(2) (1960),385-404.
[19]A. Andreotti, E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on com-plex manifolds, Inst. Hautesdes Sci. Publ. Math.25 (1965),81-130.
[20]D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, AMS (1969), Proc. Steklov Inst.90.
[21]D. Arapura, Local cohomology of sheaves of differential forms and Hodge theory, Preprint
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[23]I. N. Baker, Ch. Pommerenke, On the iteration of analytic functions in a half-plane II, J. London Math. Soc.20(2) (1979),255-258.
[24]E. Berkson, H. Porta, Hermitian operators and one-parameter groups of isometries in Hardy spaces, Trans. Amer. Math. Soc.185 (1973),331-354.
[25]B. Berndtsson, N. Sibony, The (?)-equation on a positive current, Invent.math.147(2) (2002),371-428.
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[27]P. Bourdon, J. Shapiro, Cyclic phenomena for composition oprators, Mem. Amer. Math. Soc.125 (1997), no.596.
[28]R. B. Burckel, An Introduction to Classical Complex Analysis, New York:Academic Press, 1979.
[29]L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies, 1967.
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[31]A. Candel, The harmonic measures of Lucy Garnett, Adv. Math.,176(2) (2003),187-247.
[32]J. G. Caughran, Polynomial approximation and spectral properties of composition operators on H2, India. Univ. Math. J.21 (1971),81-84.
[33]I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics,115. A-cademic Press, Inc., Orlando, FL,1984.
[34]G. Choquet, Lectures on analysis, Vol. II:Representation Theory, W. A. Benjamin,Inc., New York-Amsterdam,1969.
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[39]J. P. Demailly, Singular Hermitian metrics on positive line bundles, In:Complex Algebraic Varieties (Bayreuth,1990), pp 87-104, Lecture Notes in Math., vol.1507. Springer, Berlin 1992.
[40]B. Deroin, Laminations par varietes complexes, These de doctorat de 1'ENS Lyon (2003).
[41]T. C. Dinh, V. A. Nguyen, N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann.354 (2012) 331-376.
[42]T. C. Dinh, N. Sibony, Pull-back currents by holomorphic maps, Manuscr. Math.123 (2007),357-371.
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[49]J. E. Fornaess, N. Sibony, Unique ergodicity of harmonic currents on singular foliations of P2, Geom. Funct. Anal.,19 (2010),1334-1377.
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[51]L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal.,51 (1983), no.3,285-311.
[52]E. Ghys, Gauss Bonnet theorem for 2-dimensional foliations, J. Func. Anal.77 (1988), 51-59.
[53]E. Ghys, Holomorphic anosov systems, Inven. Math.119 (1995),585-614.
[54]D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag,1983.
[55]A. Haefliger, Some remarks on foliations with minimal leaves, J. Diff. Geo.,15(2) (1980), 269-284.
[56]R. Harvey, A.W. Knapp, Positive (p,p) forms, Wirtinger's inequality, and currents, In: Vitter Kujala, ed., Value distribution theory, Dekker, NY,1974.
[57]D. A. Herrero, Z. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J.39 (1990),819-830.
[58]C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J.41 (1974),693-710.
[59]A. Huckleberry, Subvarieties of homogeneous and almost homogeneous manifolds, In: Skoda et al, ed., Contributions to Complex Analysis and Analytic Geometry, Vieweg, Braunschweig,1994.
[60]F. Jafari, Angular derivatives in polydiscs, Indian J. Math.35 (1993),197-212.
[61]M. Jarnicki, P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, Berlin,1993.
[62]M. T. Jury, Valiron's theorem in the unit ball and spectra of composition operators, Jour. Math. Anal, and Appl.368 (2010),482-490.
[63]V. Kaimanovich, Brownian motions on foliations:entropy, invariant measures, mixing, Funct. Anal. Appl.,22 (1989) 326-328.
[64]H. Kamowitz, The spectra of composition operators on Hp, J. Funct. Anal.18 (1975), 132-150.
[65]S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York 1970.
[66]S. Kobayashi, Intrinsic distances, measures, and geometric function theory, Bull.Amer. Math. Soc.82 (1976),357-416.
[67]K. Kodaira, A differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA 39 (1953) 1268-1273,
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[2]T. Ohsawa, On projectively embeddable complex-foliated structure, Publ. Res. Inst. Math. Sci.48 (2012), no.3,735-747.
[3]J. E. Fornaess, E. F. Wold, Solving (?)b on hyperbolic laminations, preprint,2011. arX-iv:1108.2286.
[4]G. Valiron, Sur l'iteration des fonctions holomorphes dans un demi-plan, Bull. Sci. Math. 55(2) (1931),105-128.
[5]F. Bracci, G. Gentili, P. P. Corradini, Valiron's construction in higher dimension, Rev. Mat. Iber. (1) 26 (2010),57-76.
[6]L. Hormander, L2 estimates and existence theorems for the 8 operator, Acta Math.113 (1965),89-152.
[7]A. Candel, Uniformization of surface laminations, Ann.Sci.Ecole Norm.Sup.26(4) (1993), 489-516.
[8]E. Ghys, Lamination par surfaces de Riemann, In:Cerveau et al, ed., Panoramas et Syntheses, Soc. Math. France, (1999),49-95.
[9]P. S. Bourdon, Density of the polynomials in Bergman spaces, Pacific J. Math.130 (1987), 215-221.
[10]C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Ame. Math. Soc.265 (1981),69-95.
[11]Ch. Pommerenke, On asymptotic iteration of anslytic functions in the disk, Analysis 1. (1981),45-61.
[12]M. Abate, The Julia-Wolff-Cratheodory theorem in poly disks, Jour, d'Anal. Math.74 (1998),275-306.
[13]M. Abate, Horospheres and iterates of holomorphic maps, Math. Z.198 (1988),225-238.
[14]M. Abate, Common fixed points of commuting holomorphic maps, Math. Ann.283 (1989), 645-655.
[15]M. Abate, Angular derivatives in strongly pseudoconvex domains, Proc. Symp. Pure Math. 52(2) (1991),23-40.
[16]W. Abikhow, Real Analytic Theory of Teichmuller Space, Lect. Notes in Math.,820, Springer-Verlag 1980.
[17]L. Ahlfors, Lectures on Quasiconformal Mappings. Wadsworth 1987.
[18]L. Ahlfors, L. Bers, Riemann's mapping theorem for variable metrics, Math. Ann.72(2) (1960),385-404.
[19]A. Andreotti, E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on com-plex manifolds, Inst. Hautesdes Sci. Publ. Math.25 (1965),81-130.
[20]D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, AMS (1969), Proc. Steklov Inst.90.
[21]D. Arapura, Local cohomology of sheaves of differential forms and Hodge theory, Preprint
[22]B. Beauzamy, An operator on a separable Hilbert space, with many hypercyclic vectors, Studia Math.87 (1988),71-78.
[23]I. N. Baker, Ch. Pommerenke, On the iteration of analytic functions in a half-plane II, J. London Math. Soc.20(2) (1979),255-258.
[24]E. Berkson, H. Porta, Hermitian operators and one-parameter groups of isometries in Hardy spaces, Trans. Amer. Math. Soc.185 (1973),331-354.
[25]B. Berndtsson, N. Sibony, The (?)-equation on a positive current, Invent.math.147(2) (2002),371-428.
[26]C. Bonatti, Gomez-Mont X, Sur le comportement statistique des feuilles de certains feuil-letages holomorphes, Monogr. Enseign. Math.,38 (2001),15-41.
[27]P. Bourdon, J. Shapiro, Cyclic phenomena for composition oprators, Mem. Amer. Math. Soc.125 (1997), no.596.
[28]R. B. Burckel, An Introduction to Classical Complex Analysis, New York:Academic Press, 1979.
[29]L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Mathematical Studies, 1967.
[30]A. Candel, L. Conlon, Foliations,1 vol. Graduate studies in Math. AMS 1999.
[31]A. Candel, The harmonic measures of Lucy Garnett, Adv. Math.,176(2) (2003),187-247.
[32]J. G. Caughran, Polynomial approximation and spectral properties of composition operators on H2, India. Univ. Math. J.21 (1971),81-84.
[33]I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics,115. A-cademic Press, Inc., Orlando, FL,1984.
[34]G. Choquet, Lectures on analysis, Vol. II:Representation Theory, W. A. Benjamin,Inc., New York-Amsterdam,1969.
[35]J. P. R. Christensen, Topology and Borel Structure:Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, Vol.10,1974.
[36]A. Connes, A Survey of Foliations and Operator Algebras, Proc. Symp. in Pure Math.38 (1982).
[37]C. de Fabritiis, G. Gentili, On holomorphic maps which commute with hyperbolic auto-morphisms, Adv. Math.144 (1999),119-136.
[38]J. P. Demailly, Complex Analytic and Algebraic Geometry, http://www-fourier.ujf-grenoble.fr/-demailly/.
[39]J. P. Demailly, Singular Hermitian metrics on positive line bundles, In:Complex Algebraic Varieties (Bayreuth,1990), pp 87-104, Lecture Notes in Math., vol.1507. Springer, Berlin 1992.
[40]B. Deroin, Laminations par varietes complexes, These de doctorat de 1'ENS Lyon (2003).
[41]T. C. Dinh, V. A. Nguyen, N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann.354 (2012) 331-376.
[42]T. C. Dinh, N. Sibony, Pull-back currents by holomorphic maps, Manuscr. Math.123 (2007),357-371.
[43]H. Donnelly, C. Fefferman, L2-cohomology and index theorem for the Bergman metric, Ann. Math.118 (1983) 593-618.
[44]N. Dunford, J. T. Schwartz, Linear Operators, Part I. General theory, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley&Sons, Inc., New York,1988.
[45]J. Duval, N. Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79(1995)487-513.
[46]H. Federer, Geometric Measure Theory, New York, Springer-Verlag,1969.
[47]J. E. Fornaess, N. Sibony, Riemann surface laminations with singularities, J. Geom. Anal. 18(2) (2008),400-422.
[48]J. E. Fornaess, N. Sibony, Harmonic currents of finite energy and laminations, Geom. Funct. Anal.15 (2005), no.5,962-1003.
[49]J. E. Fornaess, N. Sibony, Unique ergodicity of harmonic currents on singular foliations of P2, Geom. Funct. Anal.,19 (2010),1334-1377.
[50]H. Furstenberg, Strict ergodicity and transformation of the torus, Ame. J. Math.83 (1961), 573-601.
[51]L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal.,51 (1983), no.3,285-311.
[52]E. Ghys, Gauss Bonnet theorem for 2-dimensional foliations, J. Func. Anal.77 (1988), 51-59.
[53]E. Ghys, Holomorphic anosov systems, Inven. Math.119 (1995),585-614.
[54]D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag,1983.
[55]A. Haefliger, Some remarks on foliations with minimal leaves, J. Diff. Geo.,15(2) (1980), 269-284.
[56]R. Harvey, A.W. Knapp, Positive (p,p) forms, Wirtinger's inequality, and currents, In: Vitter Kujala, ed., Value distribution theory, Dekker, NY,1974.
[57]D. A. Herrero, Z. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J.39 (1990),819-830.
[58]C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J.41 (1974),693-710.
[59]A. Huckleberry, Subvarieties of homogeneous and almost homogeneous manifolds, In: Skoda et al, ed., Contributions to Complex Analysis and Analytic Geometry, Vieweg, Braunschweig,1994.
[60]F. Jafari, Angular derivatives in polydiscs, Indian J. Math.35 (1993),197-212.
[61]M. Jarnicki, P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, Berlin,1993.
[62]M. T. Jury, Valiron's theorem in the unit ball and spectra of composition operators, Jour. Math. Anal, and Appl.368 (2010),482-490.
[63]V. Kaimanovich, Brownian motions on foliations:entropy, invariant measures, mixing, Funct. Anal. Appl.,22 (1989) 326-328.
[64]H. Kamowitz, The spectra of composition operators on Hp, J. Funct. Anal.18 (1975), 132-150.
[65]S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York 1970.
[66]S. Kobayashi, Intrinsic distances, measures, and geometric function theory, Bull.Amer. Math. Soc.82 (1976),357-416.
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