The topological classification of cascades on closed two-dimensional manifolds.

*(English. Russian original)*Zbl 0705.58038
Russ. Math. Surv. 45, No. 1, 1-35 (1990); translation from Usp. Mat. Nauk 45, No. 1(271), 3-32 (1990).

This is a review paper on the resuls, mainly by the authors, of the theory of dynamical systems with discrete times (cascades) on two- dimensional manifolds (for the review of the respective results for the systems with continuous time (flows) see e.g. the same authors in Russ. Math. Surv. 41, No.1, 183-208 (1986); translation from Usp. Mat. Nauk 41, No.1, 149-169 (1986; Zbl 0615.58015). The authors discuss mainly the problem of the topological equivalence of A-cascades [see e.g. S. Smale, Bull. Am. Math. Soc. 73, 747-817 (1967; Zbl 0202.552) for definitions].

This problem consists of two parts: the description of the complete system of topological invariants of diffeomorphisms and the construction of the standard diffeomorphism in each class of the topologically equivalent ones. The former problem is discussed in the first chapter of the paper. The authors show that this problem can be reduced to the similar problem of basis sets of nonwandering sets of diffeomorphisms. The topological classification of a diffeomorphism restriction to the basis set is given in terms of the invariants of so-called supports of the basis set. As for the second part (the construction of the standard representative), authors give its solutions basing on the Nielsen theory of the homotopic classification of the homeomorphisms of the two- dimensional manifolds (for the exposition of the Nielsen theory see e.g. J. Gilman, Adv. Math. 40, 68-96 (1981; Zbl 0474.57005).

This problem consists of two parts: the description of the complete system of topological invariants of diffeomorphisms and the construction of the standard diffeomorphism in each class of the topologically equivalent ones. The former problem is discussed in the first chapter of the paper. The authors show that this problem can be reduced to the similar problem of basis sets of nonwandering sets of diffeomorphisms. The topological classification of a diffeomorphism restriction to the basis set is given in terms of the invariants of so-called supports of the basis set. As for the second part (the construction of the standard representative), authors give its solutions basing on the Nielsen theory of the homotopic classification of the homeomorphisms of the two- dimensional manifolds (for the exposition of the Nielsen theory see e.g. J. Gilman, Adv. Math. 40, 68-96 (1981; Zbl 0474.57005).

Reviewer: L.Pastur

##### MSC:

37D99 | Dynamical systems with hyperbolic behavior |

37D15 | Morse-Smale systems |

54H20 | Topological dynamics (MSC2010) |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

28D05 | Measure-preserving transformations |