Braids and dynamics (Q2673453)

Mixing of fluids is an important example of dynamical mixing, which is relevant in many applications and industrial processes. The motivation for this book comes from the study of mixing in laminar fluids, such as dyeing inside a viscous material (or mixing ingredients in a pastry cake). Imagine that we are (gently) stirring black ink inside a shallow circular container using three moving rods and producing a ``figure-eight'' motion. Identifying the container with a disk and the rods with punctures inside it, one can model this situation with the action on a close curve around the punctures as we shuffle them using clockwise and counter-clockwise movements. To study this model, the author begins in Chapter 2 with a quick introduction to the fundamental group (and the first homology group) of the torus and its Mapping Class Group (MCG), explaining, in particular, the classification of its elements into elliptic, parabolic and hyperbolic. In Chapter 3, it is explained how, quotienting the torus by hyperelliptic involution, one gets a topological sphere with four punctures. Pulling open one of the punctures one gets back to the disk with three punctures, and so one can analyze the dynamic of the previous three-rod system using algebraic topology. In particular, this analysis shows that the presence and efficiency of mixing heavily depends on the relative motion of the rods. Another source of examples for the book is the study of taffy pullers: machines designed to stretch and fold repeatedly the taffy (a chewy candy) so that the air gets trapped inside the mix, resulting in a softer texture of the candy. Following the previous chapters, this system can be modeled as a disk \(D_n\) with \(n \) punctures (corresponding to the rods). However, taffy pullers usually have more than three rods, and so it is necessary to introduce a different formalism to study them. \textit{J. S. Birman} and \textit{T. E. Brendle} [in: Handbook of knot theory. Amsterdam: Elsevier. 19--103 (2005; Zbl 1094.57006)] have shown that the MCG of \(D_n\) can be identified with the Artin braid group \( B_n \), which are introduced in Chapter 4, where the isomorphism is mentioned but neither explicitly given nor proven. In order to manipulate braids, three representations of the group \( B_n \) are introduced here: Artin's one, the free homotopy representation and finally the Burau one, which combines the advantages of the first two. Chapter 5 explains the Nielsen-Thurston classification of elements of the MCG of finite type surfaces, with particular emphasis on pseudo-Anosov mapping classes. Contrary to the majority of treatments of this topic, the case of surfaces with boundary is treated in detail, since they are fundamental for applications. The dilation of a pseudo-Anosov map is introduced here too. For the purpose of taffy pullers, and more generally mixing in fluids, this quantity is relevant since it measures the speed of mixing. In other words, how quickly the taffy gets stretched and folded around (which geometrically corresponds to how quickly it approaches the unstable foliation of the pseudo-Anosov map). The logarithm of the dilation of a pseudo-Anosov map is equal to its topological entropy, which is defined, using Bowen balls, in Chapter 6. As a first approximation for calculating entropy, the Burau estimate is obtained using the relation with the growth of word lengths in finitely generated groups and the isomorphism between \(B_n\) and \(\operatorname{MCG}(D_n)\). Unfortunately, the Burau estimates can be very imprecise. To solve this issue, other two methods are explained in Chapters 7 and 8: the Bestvina-Handel algorithm using train tracks [\textit{M. Bestvina} and \textit{M. Handel}, Topology 34, No. 1, 109--140 (1995; Zbl 0837.57010)] and Dynnikov coordinates [\textit{I. Dynnikov} and \textit{B. Wiest}, J. Eur. Math. Soc. (JEMS) 9, No. 4, 801--840 (2007; Zbl 1187.20045)]. Train tracks were introduced by Thurston to obtain a combinatorial model of (an approximation of) the unstable foliation of a pseudo-Anosov map. The definition and properties of train tracks is given here using intuition and pictures rather than formal statements. Also the Bestvina-Handel algorithm is explained using examples, showing its power and versatility, since it allows to recover plenty of information about the pseudo-Anosov class starting with only a train track. Since for the purpose of the book the dilation is the only relevant quantity, Dynnikov coordinates on the space of multicurves on \(D_n\) are introduced as quicker and easier way to compute it. The rules used to update these coordinates under the action of the braid group \(B_n\) are obtained in the Appendix A (following unpublished notes of Spencer A.\ Smith). These rules are more simply described using the \textit{max-plus algebra}. The last two Chapters, 9 and 10, are used to introduce the \texttt{Braidlab} library, a Matlab package written by the author and Marko Budišić to define, analyze and calculate relevant quantities of braids described in the earlier chapters. Very interestingly, in the last chapter it is explained how one can produce braids from data, which is useful for a variety of problems, such as, for instance, flight of flocks of birds. A problem here is that usually real life data do not form a braid in the usual sense since the starting starting and ending position of the ``data braid'' are not the same. Some strategies for dealing with this problem are given and applied with the use of \texttt{Braidlab}. As it is evident from the above description, the book, despite its short length, covers a great deal of topics coming from algebraic topology, algebra and dynamics in low dimension. The style is not too formal, avoiding the classical ``definition-lemma-theorem'' style in favor of a much more relaxed writing. Many important results are not proven. In some cases the strategy or reasoning behind the proof is given, in some others not. However, the relevant literature is always suggested to the reader interested in the details. The (many!) excellent pictures are an essential part of the exposition. Finally, each chapter opens with a short introduction and closes with a summary of its results, with some extra insights and a teaser of what is coming after. This is especially useful since a lot of material can be packed in each chapter. This book is a very pleasant to read, linking seemingly unrelated topic like braid and fluid dynamics, perhaps a bit easier for a pure mathematician than for an applied one, but still very much enjoyable.