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Shadowing in dynamical systems. (English) Zbl 0954.37014
Lecture Notes in Mathematics. 1706. Berlin: Springer. xvii, 271 p. (1999).
The book contains a detailed and comprehensive presentation of the shadowing property of dynamical systems. While the results contained in the book are not new, the author succeeds in presenting them in a unified way. The discussion is enhanced by a fairly large collection of fundamental examples.
The basic object of consideration is a homeomorphism \(\varphi\) of a metric space \((X,r)\). This homeomorphism is said to have the pseudo orbit tracing property on a subset \(Y\) of \((X,r)\) if for a given \(\varepsilon> 0\) there exists some \(d>0\) such that for every sequence \(\{x_k\} \subset Y\) with \(r(\Phi(x_k), x_{k+1})< d\) a point \(x\in X\) can be found with \(r(\Phi^k x,x_k)< \varepsilon\) for all \(k\in \mathbb{Z}\).
Section 1 of the book introduces this and related properties of a homeomorphism. It is shown that a diffeomorphism \(\varphi\) of a manifold \(M\) has a pseudo orbit tracing property on a neighborhood of a compact invariant hyperbolic set. For this fact, two proofs are given which go back to Anosov and Bowen. Moreover the section contains a detailed discussion of an analogous property for family of \(C^1\)-maps of Banach spaces to which many results on shadowing can be reduced.
In Section 2, shadowing is related to other well known properties of topological dynamical systems. A result of Walters is presented which states that an expansive diffeomorphism of a smooth closed manifold which has the pseudo orbit tracing property is structurally stable. On the other hand, every structurally stable diffeomorphism has the pseudo orbit tracing property. It is shown that structural stability is in fact equivalent to a certain shadowing property.
Section 3 contains a large collection of specific examples which illustrate the results presented in sections 1 and 2. Among these is a complete discussion of 1-dimensional dynamical systems, linear maps and their induced action on the sphere, lattice systems and attractors for evolution systems.
Section 4 is devoted to applications of shadowing to the numerical treatment of dynamical systems.
The book is well enough written to make it a very useful reference book for the shadowing phenomenon for topological dynamical systems.

MSC:
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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