Applied symbolic dynamics and chaos.

*(English)*Zbl 0914.58017
Directions in Chaos. 7. Singapore: World Scientific. xv, 443 p. (1998).

In this book the following approach to dynamical systems is being exploited: if we introduce a finite partition of the phase space and consider the symbolic sequences obtained from labeling trajectories of points with respect to this partition, then we obtain a symbolic system from which many properties of the original system can be derived. This is what applied symbolic dynamics stands for.

Some theoretical background concerning abstract topological and symbolic dynamics is provided in the book, however it is not assumed that the reader is interested in theoretical details and several notions are introduced only intuitively. It is so with the key notion of chaos. A diffused computer plot-out must suffice as a criterion until positive entropy is mentioned later in the text. It must be admitted that in spite of this lack of formalism in most cases the argumentation is convincing and well-presented.

The longest and best handled subject is unimodal maps of the interval. The crucial role plays the kneading sequence, i.e., one obtained from the trajectory of the critical point. Moreover, compatibility conditions and the ordering rule are formulated (the first one determines if a given sequence represents an existing trajectory, the latter says which of two symbolic sequences comes from a trajectory of a smaller point in the interval). In a one-parameter family of such maps the phenomena observed on the bifurcation diagram are studied using the above tools. It is important that we can parameterize the family by kneading sequences (ordered by the ordering rule).

Similar studies are conducted for maps with many critical points. An interesting part concerns the phenomenon of symmetry breaking for periodic orbits in a one-parameter family of symmetric (odd) maps with two critical points. For two-parameter families we refer to the kneading plane (consisting of pairs of kneading sequences). Such properties as positive entropy, stability of periodic orbits, etc., can be derived from the positioning with respect to certain natural lines on the kneading plane. Maps with discontinuities are discussed, too.

In the symbolic representation of circle maps the rotation number (or the rotation interval) plays the crucial role. In the case of a map with two monotone branches the admissibility condition is based on the Farey representation of the rotation number.

The Tél map, the Lozi map, and the Hénon map are examples of two-dimensional dynamics where symbolic methods can be applied, as it is done in the book.

The next sections are devoted to the following subjects: application of symbolic dynamics to ordinary differential equations, the problem of counting periodic orbits, grammatical complexity of languages, and knot theory. Perhaps, due to high intricacy of the theoretical part, only a very general discussion is presented. Apparently, a too intuitive approach may be considered an obstacle in understanding the presented ideas. At least, one can learn about the basics of number theory, the theory of languages, and knot theory.

In the appendix, some computer routines for practical realization of the introduced methodology are provided.

The book can be recommended not only as an introduction to applied symbolic dynamics, but also as a source of motivation for studying dynamical systems from the theoretical point of view. However, as the authors themselves advise, a deeper study cannot be based exclusively on this book.

Some theoretical background concerning abstract topological and symbolic dynamics is provided in the book, however it is not assumed that the reader is interested in theoretical details and several notions are introduced only intuitively. It is so with the key notion of chaos. A diffused computer plot-out must suffice as a criterion until positive entropy is mentioned later in the text. It must be admitted that in spite of this lack of formalism in most cases the argumentation is convincing and well-presented.

The longest and best handled subject is unimodal maps of the interval. The crucial role plays the kneading sequence, i.e., one obtained from the trajectory of the critical point. Moreover, compatibility conditions and the ordering rule are formulated (the first one determines if a given sequence represents an existing trajectory, the latter says which of two symbolic sequences comes from a trajectory of a smaller point in the interval). In a one-parameter family of such maps the phenomena observed on the bifurcation diagram are studied using the above tools. It is important that we can parameterize the family by kneading sequences (ordered by the ordering rule).

Similar studies are conducted for maps with many critical points. An interesting part concerns the phenomenon of symmetry breaking for periodic orbits in a one-parameter family of symmetric (odd) maps with two critical points. For two-parameter families we refer to the kneading plane (consisting of pairs of kneading sequences). Such properties as positive entropy, stability of periodic orbits, etc., can be derived from the positioning with respect to certain natural lines on the kneading plane. Maps with discontinuities are discussed, too.

In the symbolic representation of circle maps the rotation number (or the rotation interval) plays the crucial role. In the case of a map with two monotone branches the admissibility condition is based on the Farey representation of the rotation number.

The Tél map, the Lozi map, and the Hénon map are examples of two-dimensional dynamics where symbolic methods can be applied, as it is done in the book.

The next sections are devoted to the following subjects: application of symbolic dynamics to ordinary differential equations, the problem of counting periodic orbits, grammatical complexity of languages, and knot theory. Perhaps, due to high intricacy of the theoretical part, only a very general discussion is presented. Apparently, a too intuitive approach may be considered an obstacle in understanding the presented ideas. At least, one can learn about the basics of number theory, the theory of languages, and knot theory.

In the appendix, some computer routines for practical realization of the introduced methodology are provided.

The book can be recommended not only as an introduction to applied symbolic dynamics, but also as a source of motivation for studying dynamical systems from the theoretical point of view. However, as the authors themselves advise, a deeper study cannot be based exclusively on this book.

Reviewer: Tomasz Downarowicz (Wrocław)

##### MSC:

37E99 | Low-dimensional dynamical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37N99 | Applications of dynamical systems |