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Elements of Asymptotic Geometry [[electronic resource] /] / Sergei Buyalo, Viktor Schroeder
| Elements of Asymptotic Geometry [[electronic resource] /] / Sergei Buyalo, Viktor Schroeder |
| Autore | Buyalo Sergei |
| Pubbl/distr/stampa | Zuerich, Switzerland, : European Mathematical Society Publishing House, 2007 |
| Descrizione fisica | 1 online resource (212 pages) |
| Collana | EMS Monographs in Mathematics (EMM) |
| Soggetto topico |
Differential & Riemannian geometry
Geometry Differential geometry |
| ISBN | 3-03719-536-3 |
| Classificazione | 51-xx53-xx |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-9910151936503321 |
Buyalo Sergei
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| Zuerich, Switzerland, : European Mathematical Society Publishing House, 2007 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Elements of differential geometry / Richard S. Millmaa, George D. Parker
| Elements of differential geometry / Richard S. Millmaa, George D. Parker |
| Autore | Millman, Richard S. |
| Pubbl/distr/stampa | Englewood Cliffs, N. J. : Prentice-Hall, c1977 |
| Descrizione fisica | xiv, 265 p. : ill. ; 24 cm. |
| Disciplina | 516.3602 |
| Altri autori (Persone) | Parker, George D. |
| Soggetto topico |
Curves in Euclidean space
Differential geometry Surfaces in Euclidean space |
| ISBN | 0132641437 |
| Classificazione |
AMS 53A04
AMS 53A05 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991000854479707536 |
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Millman, Richard S.
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| Englewood Cliffs, N. J. : Prentice-Hall, c1977 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. del Salento | ||
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Elements of General Relativity [[electronic resource] /] / by Piotr T. Chruściel
| Elements of General Relativity [[electronic resource] /] / by Piotr T. Chruściel |
| Autore | Chruściel Piotr T |
| Edizione | [1st ed. 2019.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2019 |
| Descrizione fisica | 1 online resource (IX, 283 p. 53 illus., 46 illus. in color.) |
| Disciplina | 530.11 |
| Collana | Compact Textbooks in Mathematics |
| Soggetto topico |
Differential geometry
Gravitation Differential Geometry Classical and Quantum Gravitation, Relativity Theory |
| ISBN | 3-030-28416-6 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introduction to tensor calculus and Riemannian geometry -- Curved space-time -- The Schwarzschild metric -- Weak fields, gravitational waves -- Stars -- Cosmology. |
| Record Nr. | UNINA-9910383831003321 |
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Chruściel Piotr T
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| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2019 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Elliptic operators, topology and asymptotic methods / John Roe
| Elliptic operators, topology and asymptotic methods / John Roe |
| Autore | Roe, John |
| Edizione | [2nd ed] |
| Pubbl/distr/stampa | Harlow : Longman, 1998 |
| Descrizione fisica | 209 p. ; 25 cm. |
| Collana | Pitman research notes in mathematics series, ISSN 02693674 ; 395 |
| Soggetto topico |
Differential geometry
Elliptic operators Global analysis (Mathematics) Index theorems |
| ISBN | 0582325021 |
| Classificazione | AMS 58G10 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991000858569707536 |
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Roe, John
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| Harlow : Longman, 1998 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. del Salento | ||
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Encyclopedia of Distances [[electronic resource] /] / by Michel Marie Deza, Elena Deza
| Encyclopedia of Distances [[electronic resource] /] / by Michel Marie Deza, Elena Deza |
| Autore | Deza Michel Marie |
| Edizione | [4th ed. 2016.] |
| Pubbl/distr/stampa | Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2016 |
| Descrizione fisica | 1 online resource (XXII, 756 p. 2 illus.) |
| Disciplina | 516.36 |
| Soggetto topico |
Differential geometry
Topology Computer mathematics Applied mathematics Engineering mathematics Differential Geometry Computational Mathematics and Numerical Analysis Mathematical and Computational Engineering |
| ISBN | 3-662-52844-4 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Part I. Mathematics of Distances: 1 General Definitions -- 2 Topological Spaces -- 3 Generalization of Metric Spaces -- 4 Metric Transforms -- 5 Metrics on Normed Structures -- Part II. Geometry and Distances: 6 Distances in Geometry -- 7 Riemannian and Hermitian Metrics -- 8 Distances on Surfaces and Knots -- 9 Distances on Convex Bodies, Cones and Simplicial Complexes -- Part III. Distances in Classical Mathematics: 10 Distances in Algebra -- 11 Distances on Strings and Permutations -- 12 Distances on Numbers, Polynomials and Matrices -- 13 Distances in Functional Analysis -- 14 Distances in Probability Theory -- Part IV. Distances in Applied Mathematics: 15 Distances in Graph Theory -- 16 Distances in Coding Theory -- 17 Distances and Similarities in Data Analysis -- 18 Distances in Systems and Mathematical Engineering -- Part V. Computer-Related Distances: 19 Distances on Real and Digital Planes -- 20 Voronoi Diagram Distances -- 21 Image and Audio Distances -- 22 Distances in Networks -- Part VI. Distances in Natural Sciences: 23 Distances in Biology -- 24 Distances in Physics and Chemistry -- 25 Distances in Earth Science and Astronomy -- 26 Distances in Cosmology and Theory of Relativity -- Part VII. Real-World Distances: 27 Length Measures and Scales -- 28 Distances in Applied Social Sciences -- 29 Other Distances. |
| Record Nr. | UNINA-9910254083903321 |
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Deza Michel Marie
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| Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2016 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Encyclopedia of Distances [[electronic resource] /] / by Michel Marie Deza, Elena Deza
| Encyclopedia of Distances [[electronic resource] /] / by Michel Marie Deza, Elena Deza |
| Autore | Deza Michel Marie |
| Edizione | [3rd ed. 2014.] |
| Pubbl/distr/stampa | Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2014 |
| Descrizione fisica | 1 online resource (731 p.) |
| Disciplina |
004
510 514 516 |
| Soggetto topico |
Geometry
Differential geometry Topology Computer mathematics Mathematics Visualization Applied mathematics Engineering mathematics Differential Geometry Computational Mathematics and Numerical Analysis Mathematical and Computational Engineering |
| ISBN | 3-662-44342-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Part I. Mathematics of Distances: 1 General Definitions -- 2 Topological Spaces -- 3 Generalization of Metric Spaces -- 4 Metric Transforms -- 5 Metrics on Normed Structures -- Part II. Geometry and Distances: 6 Distances in Geometry -- 7 Riemannian and Hermitian Metrics -- 8 Distances on Surfaces and Knots -- 9 Distances on Convex Bodies, Cones and Simplicial Complexes -- Part III. Distances in Classical Mathematics: 10 Distances in Algebra -- 11 Distances on Strings and Permutations -- 12 Distances on Numbers, Polynomials and Matrices -- 13 Distances in Functional Analysis -- 14 Distances in Probability Theory -- Part IV. Distances in Applied Mathematics: 15 Distances in Graph Theory -- 16 Distances in Coding Theory -- 17 Distances and Similarities in Data Analysis -- 18 Distances in Systems and Mathematical Engineering -- Part V. Computer-Related Distances: 19 Distances on Real and Digital Planes -- 20 Voronoi Diagram Distances -- 21 Image and Audio Distances -- 22 Distances in Networks -- Part VI. Distances in Natural Sciences: 23 Distances in Biology -- 24 Distances in Physics and Chemistry -- 25 Distances in Earth Science and Astronomy -- 26 Distances in Cosmology and Theory of Relativity -- Part VII. Real-World Distances: 27 Length Measures and Scales -- 28 Distances in Applied Social Sciences -- 29 Other Distances. |
| Record Nr. | UNINA-9910299980903321 |
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Deza Michel Marie
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| Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2014 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees [[electronic resource] ] : Applications to Non-Archimedean Diophantine Approximation / / by Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
| Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees [[electronic resource] ] : Applications to Non-Archimedean Diophantine Approximation / / by Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin |
| Autore | Broise-Alamichel Anne |
| Edizione | [1st ed. 2019.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2019 |
| Descrizione fisica | 1 online resource (viii, 413 pages) : illustrations |
| Disciplina | 516.362 |
| Collana | Progress in Mathematics |
| Soggetto topico |
Dynamics
Ergodic theory Differential geometry Group theory Number theory Convex geometry Discrete geometry Probabilities Dynamical Systems and Ergodic Theory Differential Geometry Group Theory and Generalizations Number Theory Convex and Discrete Geometry Probability Theory and Stochastic Processes |
| ISBN | 3-030-18315-7 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introduction -- Negatively curved geometry -- Potentials, critical exponents and Gibbs cocycles -- Patterson-Sullivan and Bowen-Margulis measures with potential on CAT(-1) spaces -- Symbolic dynamics of geodesic flows on trees -- Random walks on weighted graphs of groups -- Skinning measures with potential on CAT(-1) spaces -- Explicit measure computations for simplicial trees and graphs of groups -- Rate of mixing for the geodesic flow -- Equidistribution of equidistant level sets to Gibbs measures -- Equidistribution of common perpendicular arcs -- Equidistribution and counting of common perpendiculars in quotient spaces -- Geometric applications -- Fields with discrete valuations -- Bruhat-Tits trees and modular groups -- Rational point equidistribution and counting in completed function fields -- Equidistribution and counting of quadratic irrational points in non-Archimedean local fields -- Counting and equidistribution of crossratios -- Counting and equidistribution of integral representations by quadratic norm forms -- A - A weak Gibbs measure is the unique equilibrium, by J. Buzzi -- List of Symbols -- Index -- Bibliography. |
| Record Nr. | UNINA-9910364957603321 |
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Broise-Alamichel Anne
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| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2019 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Ergodic Theory and Negative Curvature [[electronic resource] ] : CIRM Jean-Morlet Chair, Fall 2013 / / edited by Boris Hasselblatt
| Ergodic Theory and Negative Curvature [[electronic resource] ] : CIRM Jean-Morlet Chair, Fall 2013 / / edited by Boris Hasselblatt |
| Edizione | [1st ed. 2017.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Springer, , 2017 |
| Descrizione fisica | 1 online resource (VII, 328 p. 68 illus., 17 illus. in color.) |
| Disciplina | 515.352 |
| Collana | Lecture Notes in Mathematics |
| Soggetto topico |
Dynamics
Ergodic theory Differential geometry Dynamical Systems and Ergodic Theory Differential Geometry |
| ISBN | 3-319-43059-9 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- 1 Introduction to Hyperbolic Dynamics and Ergodic Theory -- 1.1 Introduction -- 1.1.1 Guided Tour -- 1.1.2 Examples -- 1.1.3 Hyperbolic Dynamics -- 1.2 Historical Sketch -- 1.2.1 Homoclinic Tangles -- 1.2.2 Geodesic Flows -- 1.2.3 Boltzmann's Fundamental Postulate -- 1.2.4 Picking Up from Poincaré -- 1.2.5 Modern Hyperbolic Dynamics -- 1.3 Hyperbolic Sets: Shadowing and Expansiveness -- 1.3.1 Definitions -- 1.3.2 Invariant Cones -- 1.3.3 Shadowing, Expansiveness, Closing -- 1.3.4 Specification -- 1.3.5 Spectral Decomposition -- 1.3.6 Stability -- 1.4 The Shadowing Theorem: Stability, Symbolic Models -- 1.4.1 The Shadowing Theorem -- 1.4.2 Stability -- 1.4.3 Markov Models -- 1.5 Basic Ergodic Theory of Hyperbolic Sets -- 1.5.1 Ergodicity and Related Notions -- 1.5.2 The Hopf Argument -- 1.5.3 Mixing from the Hopf Argument -- 1.5.4 Multiple Mixing from the One-Sided Hopf Argument -- 1.6 Contractions and Invariant Manifolds -- 1.6.1 The Contraction-Mapping Principle -- 1.6.2 The Spectrum of a Linear Map -- 1.6.3 Hyperbolic Linear Maps -- 1.6.4 Stable and Unstable Manifolds of a Fixed Point -- 1.6.5 Stable and Unstable Foliations -- 1.6.6 Applications: Livschitz Theory and Local Product Structure -- 1.6.6.1 Exponential Closing -- 1.6.6.2 The Livschitz Theorem -- 1.6.6.3 Smooth Invariant Measures for Anosov Diffeomorphisms -- 1.6.6.4 Local Product Structure -- 1.7 Ergodic Theory -- 1.7.1 Asymptotic Distribution, Invariant Measures -- 1.7.2 Existence of Invariant Measures and Recurrence -- 1.7.3 The Birkhoff Ergodic Theorem -- 1.7.4 Existence of Asymptotic Distribution -- 1.7.5 The Birkhoff Ergodic Theorem for Flows -- 1.7.6 The von Neumann Mean Ergodic Theorem -- 1.7.7 Ergodicity and Unique Ergodicity -- 1.7.8 Isomorphism and Factors -- 1.7.9 Topological and Probabilistic Recurrence -- 1.7.10 Ergodicity of Translations.
1.7.10.1 First Proof of Unique Ergodicity -- 1.7.10.2 Second Proof of Unique Ergodicity -- 1.7.10.3 A Third Proof and an Application -- 1.7.11 Circle Homeomorphisms -- 1.7.12 Extensions of Rotations -- 1.7.13 Ergodicity of Expanding Maps and Toral Automorphisms -- 1.7.14 The Gauss Map -- 1.7.15 Bernoulli Shifts -- 1.7.16 Mixing -- 1.7.17 Toral Translations and Expanding Maps -- 1.7.18 Rates of Mixing and Decay of Correlations -- 1.7.19 Spectral Isomorphism and Invariants -- References -- 2 On Iteration and Asymptotic Solutions of Differential Equations by Jacques Hadamard -- 3 Dynamics of Geodesic and Horocyclic Flows -- 3.1 Introduction -- 3.1.1 First Exercises -- 3.2 Topological Dynamics of the Horocyclic Flow -- 3.2.1 Nonarithmeticity, Mixing of the Geodesic Flow, Density of Horocycles -- 3.2.2 The Horocycle (hsv) Is Dense Iff the Geodesic (gt v) Is Not Quasiminimizing -- 3.2.3 Geometrically Finite Surfaces -- 3.2.4 Some More Exercises -- 3.3 Invariant Measures for the Horocyclic Flow -- 3.3.1 The Hopf Coordinates -- 3.3.2 The Hopf Argument, Ergodicity and Mixing of the Liouville Measure -- 3.3.3 Unique Ergodicity of the Horocyclic Flow -- 3.3.4 The Finite Volume Case -- 3.3.4.1 About the Proof of Unique Ergodicity -- 3.3.4.2 Nondivergence of Horocycles -- 3.3.4.3 Conclusion of the Proof -- 3.3.5 Geometrically Finite Case -- 3.3.5.1 The Patterson-Sullivan Construction -- 3.3.5.2 The Burger-Roblin Measure -- 3.3.5.3 Equidistribution of Horocycles Towards the Bowen-Margulis Measure -- 3.3.6 Geometrically Infinite Surfaces -- 3.3.7 Exercises -- References -- 4 Ergodicity of the Weil-Petersson Geodesic Flow -- 4.1 The Proof of Ergodicity -- 4.1.1 The Ergodicity Theorem -- 4.1.2 Hyperbolic Dynamics -- 4.1.3 The Hopf Argument -- 4.1.4 Nonuniform Hyperbolicity -- 4.1.5 Addressing Singularities: The Katok-Strelcyn Criteria. 4.1.6 The Case of the Punctured Torus -- 4.2 Geodesic Flows -- 4.2.1 Vertical and Horizontal Subspaces and the Sasaki Metric -- 4.2.2 The Geodesic Flow and Jacobi Fields -- 4.2.3 Matrix Jacobi and Riccati Equations -- 4.2.4 Perpendicular Jacobi Fields and Invariant Subbundles -- 4.2.5 Consequences of Negative Curvature and Unstable Jacobi Fields -- 4.3 An Ergodicity Criterion for Incomplete Geodesic Flows -- References -- 5 Ergodicity of Geodesic Flows on Incomplete Negatively Curved Manifolds -- 5.1 Introduction -- 5.1.1 Ergodicity Criterion for a Certain Class of Geodesic Flows -- 5.1.2 Outline of Proof of Theorem 5.1.1 -- 5.1.2.1 Hopf's Argument for Anosov Systems -- 5.1.2.2 Hopf's Argument in the Context of Singular Hyperbolic Geodesic Flows -- 5.1.3 Organization of the Text -- 5.2 Geometry of Tangent Bundles -- 5.2.1 Riemannian Metrics and Curvature Tensors -- 5.2.2 The Tangent Bundle to a Tangent Bundle -- 5.3 First Derivative of Geodesic Flows and Jacobi Fields -- 5.3.1 Computation of the First Derivative of Geodesic Flows -- 5.3.2 Perpendicular Jacobi Fields and Invariant Subbundles -- 5.3.3 Matrix Jacobi and Ricatti Equations -- 5.3.4 An Estimate for the First Derivative of a Geodesic Flow -- 5.4 Hyperbolicity of Certain Geodesic Flows -- 5.5 Stable Manifolds of Certain Geodesic Flows -- 5.5.1 Local (Pesin) Stable Manifolds for Certain Geodesic Flows -- 5.5.2 Global Stable Manifolds of Certain Geodesic Flows -- 5.5.2.1 Stable Jacobi Fields and Stable Horospheres -- 5.5.2.2 Geometry of the Stable and Unstable Horospheres -- 5.5.2.3 Absolute Continuity of Global Stable Manifolds -- 5.6 Proof of Theorem 5.1.1 via Hopf's Argument -- References -- 6 The Dynamics of the Weil-Petersson Flow -- 6.1 Introduction -- 6.1.1 An Overview of the Dynamics of WP Flow -- 6.1.2 Ergodicity of WP Flow: Outline of Proof. 6.1.2.1 A Quick Review of Hopf's Argument -- 6.1.2.2 Hopf's Argument in the Context of WP Flow -- 6.1.2.3 A Brief Comment on the Verification of the Ergodicity Criterion for WP Flow -- 6.1.3 Rates of Mixing of WP Flow -- 6.1.4 Organization of the Text -- 6.2 Moduli Spaces of Riemann Surfaces and the Weil-Petersson Metric -- 6.2.1 Definition and Examples of Moduli Spaces -- 6.2.2 Teichmüller Metric -- 6.2.3 Teichmüller Spaces and Mapping-Class Groups -- 6.2.4 Fenchel-Nielsen Coordinates -- 6.2.5 Cotangent Bundle to Moduli Spaces of Riemann Surfaces -- 6.2.6 Integrable Quadratic Differentials -- 6.2.7 Teichmüller and Weil-Petersson Metrics -- 6.2.8 Ergodicity of WP Flow: Outline of Proof Revisited -- 6.3 Geometry of the Weil-Petersson Metric -- 6.3.1 Items (I) and (II) of Theorem 3 for WP Metric -- 6.3.2 Item (III) of Theorem 3 for WP Metric -- 6.3.3 Item (IV) of Theorem 3 for WP Metric -- 6.3.3.1 Wolpert's Formulas for the Curvatures of the WP Metric -- 6.3.3.2 Bounds for the First Two Derivatives of WP Metric: Overview -- 6.3.3.3 Quasi-Fuchsian Locus QF(S) and McMullen's 1-Forms θWP -- 6.3.3.4 ``Cauchy Estimate'' of ωWP After Burns-Masur-Wilkinson -- 6.3.4 Item (V) of Theorem 3 for WP Metric -- 6.3.5 Item (VI) of Theorem 3 for WP Flow -- 6.4 Decay of Correlations for the Weil-Petersson Geodesic Flow -- 6.4.1 Rates of Mixing of the WP Flow on T1Mg,n I: Proof of Theorem 11 -- 6.4.2 Rates of Mixing of the WP Flow on T1Mg,n II: Proof of Theorem 12 -- 6.4.2.1 Excursions Near the Cusp and Suspension Flows -- 6.4.2.2 Rapid Mixing of Contact Suspension Flows -- 6.4.2.3 The Derivative of the Roof Function -- 6.4.2.4 Some Estimates for the Expansion Factors Λ(β) -- References -- 7 A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature -- 7.1 Introduction -- 7.2 Arithmetic Applications. 7.2.1 Basic and Classic Diophantine Approximation -- 7.2.2 An Approximation Framework -- 7.2.3 Diophantine Approximation in R by Quadratic Irrationals -- 7.2.4 Equidistribution of Rational Points in R and C -- 7.2.5 Equidistribution and Counting in the Heisenberg Group -- 7.3 Measures in Negative Curvature -- 7.3.1 A Classical Link Between Basic Diophantine Approximation and Hyperbolic Geometry -- 7.3.2 Negative Curvature Background -- 7.3.3 The Various Measures -- 7.4 Geometric Equidistribution and Counting -- 7.4.1 Equidistribution and Counting of Common Perpendicular -- 7.4.2 Towards the Arithmetic Applications -- References. |
| Record Nr. | UNISA-996466528403316 |
| Cham : , : Springer International Publishing : , : Imprint : Springer, , 2017 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
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Ergodic Theory and Negative Curvature [[electronic resource] ] : CIRM Jean-Morlet Chair, Fall 2013 / / edited by Boris Hasselblatt
| Ergodic Theory and Negative Curvature [[electronic resource] ] : CIRM Jean-Morlet Chair, Fall 2013 / / edited by Boris Hasselblatt |
| Edizione | [1st ed. 2017.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Springer, , 2017 |
| Descrizione fisica | 1 online resource (VII, 328 p. 68 illus., 17 illus. in color.) |
| Disciplina | 515.352 |
| Collana | Lecture Notes in Mathematics |
| Soggetto topico |
Dynamics
Ergodic theory Differential geometry Dynamical Systems and Ergodic Theory Differential Geometry |
| ISBN | 3-319-43059-9 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- 1 Introduction to Hyperbolic Dynamics and Ergodic Theory -- 1.1 Introduction -- 1.1.1 Guided Tour -- 1.1.2 Examples -- 1.1.3 Hyperbolic Dynamics -- 1.2 Historical Sketch -- 1.2.1 Homoclinic Tangles -- 1.2.2 Geodesic Flows -- 1.2.3 Boltzmann's Fundamental Postulate -- 1.2.4 Picking Up from Poincaré -- 1.2.5 Modern Hyperbolic Dynamics -- 1.3 Hyperbolic Sets: Shadowing and Expansiveness -- 1.3.1 Definitions -- 1.3.2 Invariant Cones -- 1.3.3 Shadowing, Expansiveness, Closing -- 1.3.4 Specification -- 1.3.5 Spectral Decomposition -- 1.3.6 Stability -- 1.4 The Shadowing Theorem: Stability, Symbolic Models -- 1.4.1 The Shadowing Theorem -- 1.4.2 Stability -- 1.4.3 Markov Models -- 1.5 Basic Ergodic Theory of Hyperbolic Sets -- 1.5.1 Ergodicity and Related Notions -- 1.5.2 The Hopf Argument -- 1.5.3 Mixing from the Hopf Argument -- 1.5.4 Multiple Mixing from the One-Sided Hopf Argument -- 1.6 Contractions and Invariant Manifolds -- 1.6.1 The Contraction-Mapping Principle -- 1.6.2 The Spectrum of a Linear Map -- 1.6.3 Hyperbolic Linear Maps -- 1.6.4 Stable and Unstable Manifolds of a Fixed Point -- 1.6.5 Stable and Unstable Foliations -- 1.6.6 Applications: Livschitz Theory and Local Product Structure -- 1.6.6.1 Exponential Closing -- 1.6.6.2 The Livschitz Theorem -- 1.6.6.3 Smooth Invariant Measures for Anosov Diffeomorphisms -- 1.6.6.4 Local Product Structure -- 1.7 Ergodic Theory -- 1.7.1 Asymptotic Distribution, Invariant Measures -- 1.7.2 Existence of Invariant Measures and Recurrence -- 1.7.3 The Birkhoff Ergodic Theorem -- 1.7.4 Existence of Asymptotic Distribution -- 1.7.5 The Birkhoff Ergodic Theorem for Flows -- 1.7.6 The von Neumann Mean Ergodic Theorem -- 1.7.7 Ergodicity and Unique Ergodicity -- 1.7.8 Isomorphism and Factors -- 1.7.9 Topological and Probabilistic Recurrence -- 1.7.10 Ergodicity of Translations.
1.7.10.1 First Proof of Unique Ergodicity -- 1.7.10.2 Second Proof of Unique Ergodicity -- 1.7.10.3 A Third Proof and an Application -- 1.7.11 Circle Homeomorphisms -- 1.7.12 Extensions of Rotations -- 1.7.13 Ergodicity of Expanding Maps and Toral Automorphisms -- 1.7.14 The Gauss Map -- 1.7.15 Bernoulli Shifts -- 1.7.16 Mixing -- 1.7.17 Toral Translations and Expanding Maps -- 1.7.18 Rates of Mixing and Decay of Correlations -- 1.7.19 Spectral Isomorphism and Invariants -- References -- 2 On Iteration and Asymptotic Solutions of Differential Equations by Jacques Hadamard -- 3 Dynamics of Geodesic and Horocyclic Flows -- 3.1 Introduction -- 3.1.1 First Exercises -- 3.2 Topological Dynamics of the Horocyclic Flow -- 3.2.1 Nonarithmeticity, Mixing of the Geodesic Flow, Density of Horocycles -- 3.2.2 The Horocycle (hsv) Is Dense Iff the Geodesic (gt v) Is Not Quasiminimizing -- 3.2.3 Geometrically Finite Surfaces -- 3.2.4 Some More Exercises -- 3.3 Invariant Measures for the Horocyclic Flow -- 3.3.1 The Hopf Coordinates -- 3.3.2 The Hopf Argument, Ergodicity and Mixing of the Liouville Measure -- 3.3.3 Unique Ergodicity of the Horocyclic Flow -- 3.3.4 The Finite Volume Case -- 3.3.4.1 About the Proof of Unique Ergodicity -- 3.3.4.2 Nondivergence of Horocycles -- 3.3.4.3 Conclusion of the Proof -- 3.3.5 Geometrically Finite Case -- 3.3.5.1 The Patterson-Sullivan Construction -- 3.3.5.2 The Burger-Roblin Measure -- 3.3.5.3 Equidistribution of Horocycles Towards the Bowen-Margulis Measure -- 3.3.6 Geometrically Infinite Surfaces -- 3.3.7 Exercises -- References -- 4 Ergodicity of the Weil-Petersson Geodesic Flow -- 4.1 The Proof of Ergodicity -- 4.1.1 The Ergodicity Theorem -- 4.1.2 Hyperbolic Dynamics -- 4.1.3 The Hopf Argument -- 4.1.4 Nonuniform Hyperbolicity -- 4.1.5 Addressing Singularities: The Katok-Strelcyn Criteria. 4.1.6 The Case of the Punctured Torus -- 4.2 Geodesic Flows -- 4.2.1 Vertical and Horizontal Subspaces and the Sasaki Metric -- 4.2.2 The Geodesic Flow and Jacobi Fields -- 4.2.3 Matrix Jacobi and Riccati Equations -- 4.2.4 Perpendicular Jacobi Fields and Invariant Subbundles -- 4.2.5 Consequences of Negative Curvature and Unstable Jacobi Fields -- 4.3 An Ergodicity Criterion for Incomplete Geodesic Flows -- References -- 5 Ergodicity of Geodesic Flows on Incomplete Negatively Curved Manifolds -- 5.1 Introduction -- 5.1.1 Ergodicity Criterion for a Certain Class of Geodesic Flows -- 5.1.2 Outline of Proof of Theorem 5.1.1 -- 5.1.2.1 Hopf's Argument for Anosov Systems -- 5.1.2.2 Hopf's Argument in the Context of Singular Hyperbolic Geodesic Flows -- 5.1.3 Organization of the Text -- 5.2 Geometry of Tangent Bundles -- 5.2.1 Riemannian Metrics and Curvature Tensors -- 5.2.2 The Tangent Bundle to a Tangent Bundle -- 5.3 First Derivative of Geodesic Flows and Jacobi Fields -- 5.3.1 Computation of the First Derivative of Geodesic Flows -- 5.3.2 Perpendicular Jacobi Fields and Invariant Subbundles -- 5.3.3 Matrix Jacobi and Ricatti Equations -- 5.3.4 An Estimate for the First Derivative of a Geodesic Flow -- 5.4 Hyperbolicity of Certain Geodesic Flows -- 5.5 Stable Manifolds of Certain Geodesic Flows -- 5.5.1 Local (Pesin) Stable Manifolds for Certain Geodesic Flows -- 5.5.2 Global Stable Manifolds of Certain Geodesic Flows -- 5.5.2.1 Stable Jacobi Fields and Stable Horospheres -- 5.5.2.2 Geometry of the Stable and Unstable Horospheres -- 5.5.2.3 Absolute Continuity of Global Stable Manifolds -- 5.6 Proof of Theorem 5.1.1 via Hopf's Argument -- References -- 6 The Dynamics of the Weil-Petersson Flow -- 6.1 Introduction -- 6.1.1 An Overview of the Dynamics of WP Flow -- 6.1.2 Ergodicity of WP Flow: Outline of Proof. 6.1.2.1 A Quick Review of Hopf's Argument -- 6.1.2.2 Hopf's Argument in the Context of WP Flow -- 6.1.2.3 A Brief Comment on the Verification of the Ergodicity Criterion for WP Flow -- 6.1.3 Rates of Mixing of WP Flow -- 6.1.4 Organization of the Text -- 6.2 Moduli Spaces of Riemann Surfaces and the Weil-Petersson Metric -- 6.2.1 Definition and Examples of Moduli Spaces -- 6.2.2 Teichmüller Metric -- 6.2.3 Teichmüller Spaces and Mapping-Class Groups -- 6.2.4 Fenchel-Nielsen Coordinates -- 6.2.5 Cotangent Bundle to Moduli Spaces of Riemann Surfaces -- 6.2.6 Integrable Quadratic Differentials -- 6.2.7 Teichmüller and Weil-Petersson Metrics -- 6.2.8 Ergodicity of WP Flow: Outline of Proof Revisited -- 6.3 Geometry of the Weil-Petersson Metric -- 6.3.1 Items (I) and (II) of Theorem 3 for WP Metric -- 6.3.2 Item (III) of Theorem 3 for WP Metric -- 6.3.3 Item (IV) of Theorem 3 for WP Metric -- 6.3.3.1 Wolpert's Formulas for the Curvatures of the WP Metric -- 6.3.3.2 Bounds for the First Two Derivatives of WP Metric: Overview -- 6.3.3.3 Quasi-Fuchsian Locus QF(S) and McMullen's 1-Forms θWP -- 6.3.3.4 ``Cauchy Estimate'' of ωWP After Burns-Masur-Wilkinson -- 6.3.4 Item (V) of Theorem 3 for WP Metric -- 6.3.5 Item (VI) of Theorem 3 for WP Flow -- 6.4 Decay of Correlations for the Weil-Petersson Geodesic Flow -- 6.4.1 Rates of Mixing of the WP Flow on T1Mg,n I: Proof of Theorem 11 -- 6.4.2 Rates of Mixing of the WP Flow on T1Mg,n II: Proof of Theorem 12 -- 6.4.2.1 Excursions Near the Cusp and Suspension Flows -- 6.4.2.2 Rapid Mixing of Contact Suspension Flows -- 6.4.2.3 The Derivative of the Roof Function -- 6.4.2.4 Some Estimates for the Expansion Factors Λ(β) -- References -- 7 A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature -- 7.1 Introduction -- 7.2 Arithmetic Applications. 7.2.1 Basic and Classic Diophantine Approximation -- 7.2.2 An Approximation Framework -- 7.2.3 Diophantine Approximation in R by Quadratic Irrationals -- 7.2.4 Equidistribution of Rational Points in R and C -- 7.2.5 Equidistribution and Counting in the Heisenberg Group -- 7.3 Measures in Negative Curvature -- 7.3.1 A Classical Link Between Basic Diophantine Approximation and Hyperbolic Geometry -- 7.3.2 Negative Curvature Background -- 7.3.3 The Various Measures -- 7.4 Geometric Equidistribution and Counting -- 7.4.1 Equidistribution and Counting of Common Perpendicular -- 7.4.2 Towards the Arithmetic Applications -- References. |
| Record Nr. | UNINA-9910257379403321 |
| Cham : , : Springer International Publishing : , : Imprint : Springer, , 2017 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Ernst Equation and Riemann Surfaces [[electronic resource] ] : Analytical and Numerical Methods / / by Christian Klein, Olaf Richter
| Ernst Equation and Riemann Surfaces [[electronic resource] ] : Analytical and Numerical Methods / / by Christian Klein, Olaf Richter |
| Autore | Klein Christian |
| Edizione | [1st ed. 2005.] |
| Pubbl/distr/stampa | Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2005 |
| Descrizione fisica | 1 online resource (X, 249 p.) |
| Disciplina | 530.15 |
| Collana | Lecture Notes in Physics |
| Soggetto topico |
Physics
Gravitation Differential geometry Mathematical Methods in Physics Classical and Quantum Gravitation, Relativity Theory Differential Geometry |
| ISBN | 3-540-31513-6 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introduction -- The Ernst Equation -- Riemann-Hilbert Problem and Fay's Identity -- Analyticity Properties and Limiting Cases -- Boundary Value Problems and Solutions -- Hyperelliptic Theta Functions and Spectral Methods -- Physical Properties -- Open Problems -- Riemann Surfaces and Theta Functions -- Ernst Equation and Twister Theory -- Index. |
| Record Nr. | UNINA-9910144603803321 |
|
Klein Christian
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| Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2005 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
