This book provides a comprehensive introduction to the applied aspects of measurable dynamics. It addresses a wide range of different topics and perspectives, such as classical dynamical systems theory, ergodic theory, graph theory, topological dynamics, symbolic dynamics and information theory, and draws connections between them.
Measurable dynamics, that is, ergodic theory, has been traditionally considered a very abstract branch of dynamical systems theory, with hardly any practical applications. However, in the particular context of analyzing transport phenomena in real-world systems, this field has received considerable scientific interest recently; a similar statement can be made with respect to the development of computational approaches. Measurable dynamics studies time-evolving densities as opposed to single trajectories and thus is particularly suited to extracting global dynamical features. The central tool is the Perron-Frobenius operator that describes the evolution of densities under the dynamics. Based on Ulam's method, this transfer operator can be numerically approximated in terms of a stochastic matrix, which can then be used to compute invariant densities as well as almost-invariant or finite-time coherent sets. These objects are crucial for analyzing and quantifying transport in complex dynamics. Alternatively, geometric approaches study transport related to invariant manifolds (as well as their time-dependent analogues) and horseshoes. All these topics are addressed in the book. In addition to these transport-related aspects, the authors also discuss a more general setting. Here, an appropriate Markov description of the dynamics allows for linking classical dynamical systems theory not only to measurable dynamics but also to topological dynamics, symbolic dynamics, and information theory (e.g., different notions of entropy).
The book contains a lot of interesting material. To fully appreciate the broad scope of topics, aspects, and links, some background knowledge on dynamical systems is certainly beneficial. Nevertheless, at least some parts of the book are also accessible with little prior knowledge. In particular, there is a nice introductory chapter that motivates the measurable dynamics perspective. Throughout the book, the different concepts and notions are illustrated by many examples, also from real-world applications, where traditional approaches are likely to fail. Finally, complete and annotated MATLAB code is provided (e.g., for discretizing phase space and for setting up the stochastic matrix), which is very helpful if one likes to really apply the concepts described in the book.
On the down side, due to the wide variety of settings, the underlying assumptions on the respective dynamical system often differ from section to section and so does the notation. In addition, the selection of topics and the focus on specific aspects appears sometimes a bit ad hoc. For instance, many measure-theoretic notions are relegated to footnotes while less central topics (e.g., hyperbolic trajectories in nonautonomous systems and finite-time Lyapunov exponents) are described in their own section or even chapter. However, these points can only be considered minor.
Although "Applied and Computational Measurable Dynamics" has contents related to those in existing books, there is only small overlap. For instance, the highly regarded monograph by Lasota & Mackey [1] gives a very comprehensive and accessible mathematical introduction to the application of ergodic theoretical concepts for the analysis of deterministic systems. It is my favorite book in the field but it does not discuss any computational issues. Ding & Zhou [2] focus on numerical aspects of transfer operator calculations and, in particular, state convergence results. Transport-related aspects are only very briefly touched. Osipenko [3] applies symbolic dynamics to investigate complex dynamical behavior and also discusses algorithmic issues but does not address any ergodic theoretical aspects.
To summarize, "Applied and Computational Measurable Dynamics" is unique in its collection of material on the subject (of which only a small portion can be discussed in this short review). It cannot substitute for an excellent introductory monograph such as [1] but it can certainly complement it: with its wide range of topics and applied perspective on measurable dynamics, it can be a useful source not only for advanced students but also for researchers in computational dynamics as well as ocean and fluid dynamics. Personally, I find it a valuable addition to my bookshelf.
- [1] A. Lasota & M.C. Mackey: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics (2nd edition), Springer, 1998.
- [2] J. Ding & A. Zhou: Nonnegative Matrices, Positive Operators, and Applications, World Scientific, 2009.
- [3] G. Osipenko: Dynamical Systems, Graphs, and Algorithms, Springer, 2006.