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Dynamics Done with Your Bare Hands: Lecture Notes by Diana Davis, Bryce Weaver, Roland K. W. Roeder, and Pablo Lessa

Françoise Dal’Bo, François Ledrappier, and Amie Wilkinson, editors
Publisher: 
European Mathematical Society
Publication Date: 
2017
Number of Pages: 
204
Format: 
Paperback
Series: 
EMS Series of Lectures in Mathematics
Price: 
44.00
ISBN: 
9783037191682
Category: 
Proceedings
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on
05/2/2017
]

This monograph was assembled from notes written for the Undergraduate Summer School program “Boundaries and Dynamics” held at the University of Notre Dame in 2015. It consists of four chapters designed to introduce advanced undergraduates and first year graduate students to the field of dynamical systems by focusing on elementary examples, exercises and “bare hands” constructions. Each chapter has a different author and each author offers a significantly different perspective on dynamical systems.

The idea of the book is to present reasonably self-contained aspects of dynamical systems that are accessible to students without requiring an extended background development. Each author attempts to build a portrait of one area from the ground up or “with bare hands”. Furthermore, by design or chance, each of the chapters shows how the concepts of dynamical systems are interwoven and interconnected with geometry, topology, algebra and number theory. Each author also attempts to lead students to questions near the frontiers of current research.

In the first chapter Diana Davis explores dynamics on surfaces using billiard flows as a basic type of continuous dynamical system. Billiard dynamics uses an idealization of a billiard ball moving on a table with a specified velocity and reflecting from the sides of the table. Davis begins with a square billiard table to establish how simple periodic trajectories can occur and to introduce the corresponding trajectories on a flat torus. From there she moves on to rational polygons where all the table’s corner angles are rational multiples of \(\pi\), and then to more complicated surfaces including polygonal identification surfaces and surfaces tiled by regular polygons. A general theme throughout is the study of shear mappings and the associated cutting sequences on translation surfaces derived from an initial billiard table.

Next Bryce Weaver explores a discrete dynamical system on the flat torus, a system he calls the “2-1-1-1 hyperbolic toral automorphism”, more commonly known as the “cat map”. This is defined by a \(2\times 2\) matrix \(A\) with largest eigenvalue \(\lambda > 1\). He does not define or discuss hyperbolic systems in general but focuses on the complicated behavior and period orbits of that one mapping. Topological entropy (which measures the exponential rate at which nearby points separate) is directly related to a counting function \(P_n\) defined by the number of periodic points of \(A\) with period less than \(n\). Two proofs that characterize the growth rate of \(P_n\) for large \(n\) are the core of the chapter. One proof gives an explicit result: the limit of \(n P_n/\lambda^n\) exists and is equal to \(\lambda/(\lambda-1)\).

Roland Roeder focuses on complex dynamics in the third chapter, and in particular on the mapping \(p_c(z)=z^2+c\) and its iterations that give rise to the Mandelbrot set \(M\) and the Julia sets \(K_c\). His primary interest is the topology of the sets \(M\) and the Julia sets \(K_c\). The first part of his chapter looks at the Mandelbrot set “from the inside out” and considers the still open question of whether the closure of \(M_0\), the set of complex values of \(c\) for which \(p_c(z)\) has an attracting periodic orbit, is equal to \(M\) itself. He then reverses perspective to look “from the outside in” by examining the Julia sets in more detail. The chapter concludes with a fascinating application of the Fatou-Julia lemma to a question of gravitational lensing in astrophysics.

In the last chapter Pablo Lessa studies a non-deterministic dynamical system arising from a random walk on an infinite graph. The primary question here is: for which infinite graphs is the walk recurrent and so visits every vertex infinitely many times almost surely? George Pólya proved that random walks on the \(\mathbb{Z}^2\) lattice are recurrent, but not on \(\mathbb{Z}^d\) for dimensions \(d\geq 3\).. It turns out that simple random walk behaves the same way (recurrent or not) for all Cayley graphs associated with a given group \(G\). The only thing that matters is the growth function of the group \(G\). Only groups with growth functions that are bounded by a quadratic polynomial in the dimension of the space lead to recurrent random walks. (This can happen only if the group \(G\) is finite or has a subgroup of finite index isomorphic to either \(\mathbb{Z}\) or \(\mathbb{Z}^2\).)

Each of the chapters has oodles of cool ideas that could interest students equipped with enough enthusiasm and a sufficient background in analysis, topology and algebra. However, most every chapter moves around a lot, from idea to idea and example to example, and it’s easy to lose track of the overall direction. While the approach may have worked well in a lecture format, it seems to have trouble keeping focus in written form. At the same time, there is a lot of excellent material here that could offer useful starting points for student projects.


Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.