Dynamics in one dimension
Let \(I\) be an interval, mostly compact, and let \(f: I\to I\) be a map. For a given point \(x\in I\) we define its orbit to be a sequence: \(x_ 0=x\) and \(x_{n+1}=f(x_ n)\). Studying the behavior of orbits is a subject of the theory of one dimensional dynamical systems. The book under review deals with such systems, and the main assumption is that the functions \(f\) defining the systems are continuous. So, ergodic properties or unimodal maps are not discussed. Yet, the continuity itself in spite of being quite general assumption is fairly good for getting deep and sophisticated results and the book justifies this claim. It covers, among others, the following topics: Sharkovskij's theorem and its extensions, turbulence, symbolic dynamics, unstable manifolds, homoclinic, limit, recurrent and non-wandering points, chain recurrence (shadowing is not discussed), minimal sets, stability and asymptotic stability, topological transitivity, and mixing and topological entropy. The last chapter is a survey of the maps of the circle. The Leitmotiv of the book is chaos. Being aware of the existence of many definitions of chaos, the authors state one more. Namely, a map \(f\) is chaotic if there exists a periodic point with a period which is not a power of 2. Then, they prove that this condition is equivalent to each of the following: (a) Some iteration \(f^ n\) of \(f\) is turbulent. (b) Some iteration \(f^ m\) of \(f\) is strictly turbulent. (c) \(f\) has a homoclinic point. (d) \(f\) has a non-wandering point which is also homoclinic. (e) There exists a point such that its limit set properly contains a periodic orbit. (f) \(f\) has a non-wandering point which has a finite orbit but is not periodic. (g) There exists a homoclinic point in the sense of Poincaré. (h) There exists an infinite set \(X\) invariant under some iteration \(f^ n\) such that the restriction of \(f^ n\) to \(X\) is topologically mixing. (i) The topological entropy of \(f\) is positive. The book may serve as a good reference source not only for one- dimensional dynamics. Many results hold true also in general situation, namely for discrete dynamical systems in compact metric spaces and the authors indicate them. The presentation of the material is clear, most proofs are complete. References contains 137 items and rich bibliographical comments all over the book are provided.
scientific article; zbMATH DE number 52876