Topological classification of structurally stable 3-diffeomorphisms with two-dimensional basis sets

2015 ◽  
Vol 97 (1-2) ◽  
pp. 304-306
Author(s):  
V. Z. Grines ◽  
Yu. A. Levchenko ◽  
O. V. Pochinka
Keyword(s):  
Topological Classification ◽  
Basis Sets ◽  
Two Dimensional ◽  
Structurally Stable

scholarly journals The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor

2020 ◽  
Vol 16 (4) ◽  
pp. 595-606
Author(s):  
V.Z. Grines ◽  
◽  
E.V. Kruglov ◽  
O.V. Pochinka ◽  
◽  
...  
Keyword(s):  
Periodic Points ◽  
Topological Classification ◽  
Basis Sets ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Geometric Invariants ◽  
Structurally Stable ◽  
Nonwandering Set

This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional attractor and finitely many isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple pairwise nonintersecting arcs. The classification of one-dimensional basis sets on surfaces has been exhaustively obtained in papers by V. Grines. He also obtained a classification of some classes of structurally stable diffeomorphisms of surfaces using combined algebra-geometric invariants. In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic differentiating invariants.

The topological classification of structurally stable 3-diffeomorphisms with two-dimensional basic sets

Nonlinearity ◽  
2015 ◽  
Vol 28 (11) ◽  
pp. 4081-4102 ◽  
Author(s):  
V Grines ◽  
Yu Levchenko ◽  
V Medvedev ◽  
O Pochinka
Keyword(s):  
Topological Classification ◽  
Two Dimensional ◽  
Structurally Stable ◽  
Basic Sets

scholarly journals On periodic mapping data of a two-dimensional torus with one saddle orbit

2019 ◽  
Vol 21 (2) ◽  
pp. 164-174
Author(s):  
Anna A. Bosova ◽  
Olga V. Pochinka
Keyword(s):  
Periodic Orbit ◽  
Nodal Point ◽  
Zeta Functions ◽  
Topological Classification ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Periodic Surface ◽  
Periodic Data ◽  
Surface Diffeomorphism

Periodic data of diffeomorphisms with regular dynamics on surfaces were studied using zeta functions in a series of already classical works by such authors as P. Blanchard, J. Franks, S. Narasimhan, S. Batterson and others. The description of periodic data for gradient-like diffeomorphisms of surfaces were given in the work of A. Bezdenezhnykh and V. Grines by means of the classification of periodic surface transformations obtained by J. Nielsen. V. Grines, O. Pochinka, S. Van Strien showed that the topological classification of arbitrary Morse-Smale diffeomorphisms on surfaces is based on the problem of calculating periodic data of diffeomorphisms with a single saddle periodic orbit. Namely, the construction of filtering for Morse-Smale diffeomorphisms makes it possible to reduce the problem of studying periodic surface diffeomorphism data to the problem of calculating periodic diffeomorphism data with a single saddle periodic orbit. T. Medvedev, E. Nozdrinova, O. Pochinka solved this problem in a general formulation, that is, the periods of source orbits are calculated from a known period of the sink and saddle orbits. However, these formulas do not allow to determine the feasibility of the obtained periodic data on the surface of this kind. In an exhaustive way, the realizability problem is solved only on a sphere. In this paper we establish a complete list of periodic data of diffeomorphisms of a two-dimensional torus with one saddle orbit, provided that at least one nodal point of the map is fixed.

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