Topological theory of dynamical sytems. Recent advances.

*(English)*Zbl 0798.54047
North-Holland Mathematical Library. 52. Amsterdam: North-Holland. viii, 416 p. (1994).

In the sixties Anosov and Smale found out close connection between structural stability of a dynamical system and its hyperbolic behaviour. Since that time the theory of differentiable dynamical systems has mostly concentrated on Anosov diffeomorphisms, Axiom A diffeomorphisms, expanding maps, their generalizations as well as on their continuous counterparts (vector fields and flows). One has discovered a number of basic properties of such systems, e.g. spectral decomposition of the nonwandering set into basic sets, existence of local canonical coordinates and Markov partitions, close connection to symbolic dynamics, expansiveness, existence of interesting invariant measures. In the middle seventies Bowen paid his attention to another characteristic property of Axiom A diffeomorphisms, so-called, the pseudo orbit tracing property (shadowing). In fact, this property had been known earlier to Anosov and some other Russian dynamists. Bowen, however, considered the shadowing to be one of the most important dynamical properties and his famous results obtained via shadowing techniques fully justified such a claim. In 1978 Walters showed that shadowing combined with expansiveness implies topological stability of a homeomorphism on a compact metric space. He also proved that a subshift is of finite type if and only if it has shadowing. The Walters paper originated the studies on topological foundations of differentiable dynamical systems on their hyperbolic sets and the book under review sums up and completes these investigations.

The book is devoted to the theory of topological Anosov homeomorphisms (TA-homeomorphisms), TA-maps and related discrete dynamical systems. We do stress that the non-invertible case is the main issue in the book. Let \((X,d)\) be a compact metric space, \(f: X\to X\) a homeomorphism. Let \(f^ n\) be the \(n\)th iteration of \(f\), \(n\in Z\). We say that \(f\) is expansive if there exists a constant \(e>0\) such that for any \(x,y\in X\), \(x\neq y\), one can find an integer \(n\) such that \(d(f^ n x,f^ n y)>e\). Let \(\delta>0\) be fixed. A sequence \(\{x_ n\}_{n\in Z}\) is a \(\delta\)- pseudo orbit if \(d(fx_ n, x_{n+1})< \delta\), for all \(n\in Z\). We say that \(f\) has the pseudo orbit tracing property if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that every \(\delta\)-pseudo orbit is \(\varepsilon\)-shadowed by some point \(x\in X\), i.e. \(d(f^ n x,x_ n)<\varepsilon\), for all \(n\in Z\). We say that \(f\) is a TA-Anosov homeomorphism if it is expansive and has the pseudo orbit tracing property.

If \(f\) is not bijective one considers \(c\)-expansiveness. We say that \(f\) is \(c\)-expansive if there exists a constant \(e>0\) such that for any two different sequences \(\{x_ n\}_{n\in Z}\), \(\{y_ n\}_{n\in Z}\) satisfying \(fx_ n= x_{n+1}\), \(fy_ n= y_{n+1}\) for all \(n\in Z\) there exists an integer \(n\) such that \(d(x_ n, y_ n)>e\). The pseudo orbit tracing property for maps is defined as for a homeomorphism above but we consider positive orbits and pseudo orbits. A continuous surjection is said to be a TA-map if it is \(c\)-expansive and has the pseudo orbit tracing property. It appears that TA-maps (TA- homeomorphisms) recover most dynamic properties known in differentiable hyperbolic case: especially those mentioned in the first paragraph. The book contains full presentation of these topics.

Another interesting problem discussed in the book is a full classification of TA-covering maps on tori up to topological conjugacy. The task is considerably complex and to work it out the authors prepare suitable tools such as local product structures and foliations, fundamental groups, universal covering spaces, inverse limit systems, solenoidal groups and fixed point index theory.

The book is intended to use by interested graduate students and working mathematicians and can be considered as a monograph. On the other hand, even though most of the main topics covered by the book are relatively new, it is not a collection of research papers like some monographs are. It is a good and useful textbook, anyway.

The book is devoted to the theory of topological Anosov homeomorphisms (TA-homeomorphisms), TA-maps and related discrete dynamical systems. We do stress that the non-invertible case is the main issue in the book. Let \((X,d)\) be a compact metric space, \(f: X\to X\) a homeomorphism. Let \(f^ n\) be the \(n\)th iteration of \(f\), \(n\in Z\). We say that \(f\) is expansive if there exists a constant \(e>0\) such that for any \(x,y\in X\), \(x\neq y\), one can find an integer \(n\) such that \(d(f^ n x,f^ n y)>e\). Let \(\delta>0\) be fixed. A sequence \(\{x_ n\}_{n\in Z}\) is a \(\delta\)- pseudo orbit if \(d(fx_ n, x_{n+1})< \delta\), for all \(n\in Z\). We say that \(f\) has the pseudo orbit tracing property if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that every \(\delta\)-pseudo orbit is \(\varepsilon\)-shadowed by some point \(x\in X\), i.e. \(d(f^ n x,x_ n)<\varepsilon\), for all \(n\in Z\). We say that \(f\) is a TA-Anosov homeomorphism if it is expansive and has the pseudo orbit tracing property.

If \(f\) is not bijective one considers \(c\)-expansiveness. We say that \(f\) is \(c\)-expansive if there exists a constant \(e>0\) such that for any two different sequences \(\{x_ n\}_{n\in Z}\), \(\{y_ n\}_{n\in Z}\) satisfying \(fx_ n= x_{n+1}\), \(fy_ n= y_{n+1}\) for all \(n\in Z\) there exists an integer \(n\) such that \(d(x_ n, y_ n)>e\). The pseudo orbit tracing property for maps is defined as for a homeomorphism above but we consider positive orbits and pseudo orbits. A continuous surjection is said to be a TA-map if it is \(c\)-expansive and has the pseudo orbit tracing property. It appears that TA-maps (TA- homeomorphisms) recover most dynamic properties known in differentiable hyperbolic case: especially those mentioned in the first paragraph. The book contains full presentation of these topics.

Another interesting problem discussed in the book is a full classification of TA-covering maps on tori up to topological conjugacy. The task is considerably complex and to work it out the authors prepare suitable tools such as local product structures and foliations, fundamental groups, universal covering spaces, inverse limit systems, solenoidal groups and fixed point index theory.

The book is intended to use by interested graduate students and working mathematicians and can be considered as a monograph. On the other hand, even though most of the main topics covered by the book are relatively new, it is not a collection of research papers like some monographs are. It is a good and useful textbook, anyway.

Reviewer: J.Ombach (Kraków)