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Geodesic and Horocyclic Trajectories / by Françoise Dal’Bo
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Universitext | SpringerLink BücherPublisher: London : Springer-Verlag London Limited, 2011Description: Online-Ressource (XII, 176p. 110 illus, digital)ISBN:- 9780857290731
- 515.39
- 515.48
- QA313
Contents:
Summary: Dynamics of Fuchsian groups -- Examples of Fuchsian Groups -- Topological dynamics of the geodesic flow -- Schottky groups -- Topological dynamics -- The Lorentzian point of view -- Trajectories and Diophantine approximations.Summary: During the past thirty years, strong relationships have interwoven the fields of dynamical systems, linear algebra and number theory. This rapport between different areas of mathematics has enabled the resolution of some important conjectures and has in fact given birth to new ones. This book sheds light on these relationships and their applications in an elementary setting, by showing that the study of curves on a surface can lead to orbits of a linear group or even to continued fraction expansions of real numbers. Geodesic and Horocyclic Trajectories presents an introduction to the topological dynamics of two classical flows associated with surfaces of curvature −1, namely the geodesic and horocycle flows. Written primarily with the idea of highlighting, in a relatively elementary framework, the existence of gateways between some mathematical fields, and the advantages of using them, historical aspects of this field are not addressed and most of the references are reserved until the end of each chapter in the Comments section. Topics within the text cover geometry, and examples, of Fuchsian groups; topological dynamics of the geodesic flow; Schottky groups; the Lorentzian point of view and Trajectories and Diophantine approximations. This book will appeal to those with a basic knowledge of differential geometry including graduate students and experts with a general interest in the area Françoise Dal’Bo is a professor of mathematics at the University of Rennes. Her research studies topological and metric dynamical systems in negative curvature and their applications especially to the areas of number theory and linear actions.PPN: PPN: 1650614292Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface""; ""Why have we undertaken this project?""; ""Who does this book address?""; ""In what spirit was the text written?""; ""What does the text cover?""; ""Contents""; ""Dynamics of Fuchsian groups""; ""Introduction to the planar hyperbolic geometry""; ""Geodesics and distance""; ""Compactification of H""; ""Hyperbolic triangles and circles""; ""Horocycles and Busemann cocycles""; ""The Poincaré disk""; ""Positive isometries and Fuchsian groups""; ""Decompositions of the group G""; ""The dynamics of positive isometries""; ""Fuchsian Groups and Dirichlet domains""
""Limit points of Fuchsian groups""""Limit set""; ""Horocyclic, conical and parabolic points""; ""Geometric finiteness""; ""Geometric finiteness and Dirichlet domains""; ""Geometric finiteness and limit points""; ""Proof of (ii) => (i) in Theorem 4.13.""; ""Proof of (i) => (iii) in Theorem 4.13.""; ""Comments""; ""Examples of Fuchsian groups""; ""Schottky groups""; ""Dynamics of Schottky groups""; ""Limit set""; ""Encoding the limit set of a Schottky group""; ""The modular group and two subgroups""; ""The modular group""; ""Congruence modulo 2 subgroup and the commutator subgroup""
""The congruence modulo 2 subgroup.""""The commutator subgroup.""; ""Expansions of continued fractions""; ""Geometric interpretation of continued fraction expansions""; ""Application to the hyperbolic isometries of the modular group""; ""Comments""; ""Topological dynamics of the geodesic flow""; ""Preliminaries on the geodesic flow""; ""The geodesic flow on T1H""; ""The geodesic flow on a quotient""; ""Topological properties of geodesic trajectories""; ""Characterization of the wandering and divergent points""; ""Applications to geometrically finite groups""
""Periodic trajectories and their periods""""Density of periodic trajectories""; ""Length spectrum""; ""Dense trajectories""; ""Comments""; ""Schottky groups and symbolic dynamics""; ""Coding""; ""The density of periodic and dense trajectories""; ""An alternate proof of Theorem III.3.3""; ""A proof of Theorem III.3.3 using symbolic dynamics""; ""An alternate proof of Theorem III.4.2""; ""A proof of Theorem III.4.2 using symbolic dynamics""; ""Applications to the general case""; ""Comments""; ""Topological dynamics of the horocycle flow""; ""Preliminaries""; ""The horocycle flow on T1 H""
""A vectorial point of view on the space of trajectories of hR""""The horocycle flow on a quotient""; ""A vectorial point of view on hR""; ""Characterization of the non-wandering set""; ""Dense and periodic trajectories""; ""Dense horocyclic trajectories""; ""Relationship between hR and gR, and application""; ""Proof of Theorem III.4.3""; ""Periodic horocyclic trajectories and their periods""; ""Geometrically finite Fuchsian groups""; ""Comments""; ""The Lorentzian point of view""; ""The hyperboloid model""; ""Construction of the metric and compactification""
""Classification of positive isometries and Busemann cocycles""
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