Inner maps: topological invariants and their applications. (Внутренние отображения: топологические инварианты и их приложения.) (Russian) Zbl 1324.37003
Pratsi Instytutu Matematyky Natsional’noï Akademiï Nauk Ukraïny. Matematyka ta ïï Zastosuvannya 101. Kyïv: Instytut Matematyky NAN Ukraïny (ISBN 978-966-02-7379-5). 225 p. (2014).
The book is devoted to the topological theory of dynamical systems generated by Trokhimchuk inner mappings, which are continuous open isolated endomorphisms of metric spaces. The developed theory allows attracting analogue of the theory of dynamical systems for the study of the fine structure of branched coverings of surfaces and other inner mappings. A classification of the inner mappings of a metric space up to topological conjugacy is proposed.
The content of the book is as follows. In the second chapter, the definition of the class of inner mappings under considerations as well as basic the topological properties of the inner mappings which are necessary for further exploitation, are described. In the third chapter “Dynamics of trajectories”, the invariant sets of recurrent points are defined for inner mappings. The properties of these invariant sets are investigated. The fourth chapter “Dynamics of neighbourhoods” is concerned with the basic topological invariants of dynamical systems generated by inner mappings related with local dynamics of mappings in a neighbourhood of a point. The conditions which determine inner mappings that have dynamical characteristics similar to branched coverings of compact closed manifolds are derived. In the fifth chapter “Semiconjugate to inner mappings homeomorphisms”, properties of factor spaces of dynamical systems such that inner mappings induce homeomorphisms are investigated. The sixth chapter “Chain-recurrent sets and main theorem of dynamics of inner mappings” deals with the features of the theory of chain-recurrent sets in relation with inner mappings.
The content of the book is as follows. In the second chapter, the definition of the class of inner mappings under considerations as well as basic the topological properties of the inner mappings which are necessary for further exploitation, are described. In the third chapter “Dynamics of trajectories”, the invariant sets of recurrent points are defined for inner mappings. The properties of these invariant sets are investigated. The fourth chapter “Dynamics of neighbourhoods” is concerned with the basic topological invariants of dynamical systems generated by inner mappings related with local dynamics of mappings in a neighbourhood of a point. The conditions which determine inner mappings that have dynamical characteristics similar to branched coverings of compact closed manifolds are derived. In the fifth chapter “Semiconjugate to inner mappings homeomorphisms”, properties of factor spaces of dynamical systems such that inner mappings induce homeomorphisms are investigated. The sixth chapter “Chain-recurrent sets and main theorem of dynamics of inner mappings” deals with the features of the theory of chain-recurrent sets in relation with inner mappings.
Reviewer: Mikhail P. Moklyachuk (Kyïv)
MSC:
| 37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 57N35 | Embeddings and immersions in topological manifolds |
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
| 57R22 | Topology of vector bundles and fiber bundles |
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 30C55 | General theory of univalent and multivalent functions of one complex variable |
