Braids and dynamics / / Jean-Luc Thiffeault

• Record Nr.: UNISA996490347603316
• Autore: Thiffeault Jean-Luc
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031047909; 9783031047893
• Descrizione fisica: 1 online resource (147 pages)
• Collana: Frontiers in applied dynamical systems ; ; Volume 9
• Disciplina: 514.224
• Soggetti: Braid theory; Fluid dynamics; Dynamics; Dinàmica de fluids; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- List of Symbols -- 1 Introduction -- 1.1 Motivation: Fluid Mixing -- 1.2 Taffy Pullers -- 1.3 Outline -- 2 Topological Dynamics on the Torus -- 2.1 Diffeomorphisms of the Torus -- 2.2 The Fundamental Group of a Surface -- 2.3 The Mapping Class Group of the Torus -- 2.4 Classification of MCG(T2) -- 2.4.1 Elliptic Case -- 2.4.2 Parabolic Case -- 2.4.3 Hyperbolic Case -- 2.5 Summary -- 3 Stretching with Three Rods -- 3.1 From the Torus to the Sphere -- 3.2 The Mapping Class Groups of S4 and D3 -- 3.3 Dehn Twists -- 3.4 Fluid Stirring with Three Rods -- 3.5 Taffy Pulling with Three Rods -- 3.6 Summary -- 4 Braids -- 4.1 Braids as Particle Dances -- 4.2 Algebraic Braids -- 4.3 Artin's Representation -- 4.4 Free Homotopy Representation -- 4.5 The Burau Representation -- 4.6 Summary -- 5 The Thurston-Nielsen Classification -- 5.1 Classification of Diffeomorphisms of a Surface -- 5.2 Pseudo-Anosov Maps -- 5.3 The Degree of the Dilatation -- 5.4 Summary -- 6 Topological Entropy -- 6.1 Definition -- 6.2 Word Length Growth -- 6.3 The Burau Estimate for the Dilatation -- 6.4 An Upper Bound -- 6.5 Summary -- 7 Train Tracks -- 7.1 The Figure-Eight Stirring Device -- 7.2 A Second Pseudo-; Anosov Example -- 7.3 A Reducible Example -- 7.4 Finding Cancellations -- 7.5 Summary -- 8 Dynnikov Coordinates -- 8.1 Coordinates for Multicurves -- 8.2 Action of Braids on Dynnikov Coordinates (Update Rules) -- 8.2.1 Update Rules for sigmai -- 8.2.2 Update Rules for sigmaiinv -- 8.3 Max-Plus Algebra -- 8.4 Mapping Classes and Dynnikov Coordinates -- 8.4.1 Finite-Order Case -- 8.4.2 Reducible Case -- 8.4.3 Pseudo-Anosov Case -- 8.5 The Word Problem -- 8.6 Summary -- 9 The Braidlab Library -- 9.1 Setup and Getting Help -- 9.2 Braids -- 9.2.1 Basic Operations -- 9.2.2 Representation and Invariants -- 9.3 Loops -- 9.3.1 Acting on Loops with Braids.; 9.3.2 Loop Coordinates for a Braid -- 9.4 Entropy and Train Tracks -- 9.4.1 Topological Entropy and Complexity -- 9.4.2 Train Track Map and Transition Matrix -- 9.5 Summary -- 10 Braids and Data Analysis -- 10.1 Braids from Closed Trajectories -- 10.1.1 Constructing a Braid from Orbit Data -- 10.1.2 An Example: Taffy Pullers -- 10.1.3 Changing the Projection Line -- 10.2 Braids from Non-closed Trajectories -- 10.2.1 Constructing a Braid from Data: An Example -- 10.2.2 Changing the Projection Line and Enforcing Closure -- 10.2.3 Finite-Time Braiding Exponent (FTBE) -- 10.3 Summary -- Derivation of Dynnikov Update Rules (Spencer A. Smith) -- A.1 Dynnikov Coordinates -- A.2 Whitehead Moves -- A.2.1 Triangulation Coordinates -- A.2.2 Whitehead Move and Update Rule -- A.3 Deriving the Update Rules -- A.3.1 Counterclockwise Switch -- A.3.2 Equivalence of Update Rules -- A.3.3 Clockwise Switch -- A.3.4 Edge Cases -- References -- Index.