Dynamics in one dimension
type: Item
instance of: scholarly article
title: Dynamics in one dimension
zbMATH Open document ID: 0746.58007
author: L. Block
author: William Andrew Coppel
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: 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 ser…
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37Axx
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 54C70
zbMATH DE Number: 52876
zbMATH Keywords: one-dimensional dynamics
zbMATH Keywords: chaos
zbMATH Keywords: entropy
zbMATH Keywords: topological dynamics
zbMATH Keywords: bibliography
reviewed by: Jerzy Ombach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0084762
OpenAlex ID: W1569683457
DOI: 10.1007/BFB0084762
description: scientific article; zbMATH DE number 52876