LEADER 04437nam 2200529 450 001 996490347603316 005 20230717155521.0 010 $a9783031047909$b(electronic bk.) 010 $z9783031047893 035 $a(MiAaPQ)EBC7080721 035 $a(Au-PeEL)EBL7080721 035 $a(CKB)24782712200041 035 $a(PPN)264956915 035 $a(EXLCZ)9924782712200041 100 $a20230202d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aBraids and dynamics /$fJean-Luc Thiffeault 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (147 pages) 225 1 $aFrontiers in applied dynamical systems ;$vVolume 9 311 08$aPrint version: Thiffeault, Jean-Luc Braids and Dynamics Cham : Springer International Publishing AG,c2022 9783031047893 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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. 327 $a9.3.2 Loop Coordinates for a Braid -- 9.4 Entropy and Train Tracks -- 9.4.1 Topological Entropy and Complexity -- 9.4.2 Train Track Map and Transition Matrix -- 9.5 Summary -- 10 Braids and Data Analysis -- 10.1 Braids from Closed Trajectories -- 10.1.1 Constructing a Braid from Orbit Data -- 10.1.2 An Example: Taffy Pullers -- 10.1.3 Changing the Projection Line -- 10.2 Braids from Non-closed Trajectories -- 10.2.1 Constructing a Braid from Data: An Example -- 10.2.2 Changing the Projection Line and Enforcing Closure -- 10.2.3 Finite-Time Braiding Exponent (FTBE) -- 10.3 Summary -- Derivation of Dynnikov Update Rules (Spencer A. Smith) -- A.1 Dynnikov Coordinates -- A.2 Whitehead Moves -- A.2.1 Triangulation Coordinates -- A.2.2 Whitehead Move and Update Rule -- A.3 Deriving the Update Rules -- A.3.1 Counterclockwise Switch -- A.3.2 Equivalence of Update Rules -- A.3.3 Clockwise Switch -- A.3.4 Edge Cases -- References -- Index. 410 0$aFrontiers in applied dynamical systems ;$vVolume 9. 606 $aBraid theory 606 $aFluid dynamics 606 $aDynamics 606 $aDinàmica de fluids$2thub 608 $aLlibres electṛnics$2thub 615 0$aBraid theory. 615 0$aFluid dynamics. 615 0$aDynamics. 615 7$aDinàmica de fluids 676 $a514.224 700 $aThiffeault$b Jean-Luc$01257836 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996490347603316 996 $aBraids and Dynamics$92914741 997 $aUNISA