This monograph uses braids to explore dynamics on surfaces, with an eye towards applications to mixing in fluids. The text uses the particular example of taffy pulling devices to represent pseudo-Anosov maps in practice. In addition, its final chapters also briefly discuss current applications in the emerging field of analyzing braids created from trajectory data. While written with beginning graduate students, advanced undergraduates, or practicing applied mathematicians in mind, the book is also suitable for pure mathematicians seeking real-world examples. Readers can benefit from some knowledge of homotopy and homology groups, but these concepts are briefly reviewed. Some familiarity with Matlab is also helpful for the computational examples.
Author(s): Jean-Luc Thiffeault
Series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 9
Publisher: Springer
Year: 2022
Language: English
Pages: 146
City: Cham
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
