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.

scholarly journals On periodic points of torus endomorphisms

2019 ◽  
Vol 21 (4) ◽  
pp. 480-487
Author(s):  
Evgeniy D. Kurenkov ◽  
Dmitriy I. Mints
Keyword(s):  
Characteristic Polynomial ◽  
Periodic Points ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Structurally Stable

It is a well-known fact that Anosov endomorphisms of n-torus which are different from automorphisms and expanding endomorphisms are not structurally stable and, in general, are not conjugated to algebraic endomorphisms. Nevertheless, hyperbolic algebraic endomorphisms of torus are conjugated with their C1 perturbations on the set of periodic points. Therefore the study of algebraic toral endomorphisms is very important. This paper is devoted to study of the structure of the sets of periodic and pre-periodic points of algebraic toral endomorphisms. Various group properties of this sets of points are studied. The density of periodic points for algebraic endomorphisms of n-torus is proved; it is clarifief how the number of periodic and pre-periodic points with a fixed denominator depends on the properties of the characteristic polynomial. The Theorem 1.1 is the main result of this paper. It contains an algorithm that allows to split the sets of periodic and pre-periodic points of a given algebraic endomorphism of two-dimensional torus.

scholarly journals Cantor Type Basic Sets of Surface $A$-endomorphisms

2021 ◽  
Vol 17 (3) ◽  
pp. 335-345
Author(s):  
V. Z. Grines ◽  
◽  
E. V. Zhuzhoma ◽  
Keyword(s):  
Closed Surface ◽  
Orientable Surface ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Closed Orientable Surface ◽  
One Dimensional ◽  
Basic Set ◽  
Basic Sets ◽  
Nonwandering Set

The paper is devoted to an investigation of the genus of an orientable closed surface $M^{2}$ which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller $\Lambda_{r}$ with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if $M^{2}$ is a torus or a sphere, then $M^{2}$ admits such an endomorphism. We also show that, if $\Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^{2}\to M^{2}$ of a closed orientable surface $M^{2}$ and $f$ is not a diffeomorphism, then $\Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^{2}\to M^{2}$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_{r}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_{r}$ is regular, then $M^{2}$ is a two-dimensional torus $\mathbb{T}^{2}$ or a two-dimensional sphere $\mathbb{S}^{2}$.

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.

scholarly journals On non-hyperbolic algebraic automorphisms of a two-dimensional torus

2021 ◽  
Vol 23 (3) ◽  
pp. 295-307
Author(s):  
Sergey V. Sidorov ◽  
Ekaterina E. Chilina
Keyword(s):  
Sufficient Conditions ◽  
Complete Classification ◽  
Orientable Surface ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Algebraic Automorphism ◽  
Necessary And Sufficient

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

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

Invariants of weak equivalence in primitive matrices

2000 ◽  
Vol 20 (2) ◽  
pp. 611-626 ◽  
Author(s):  
RICHARD SWANSON ◽  
HANS VOLKMER
Keyword(s):  
Inverse Limit ◽  
Direct Application ◽  
Topological Classification ◽  
Weak Equivalence ◽  
Transition Matrices ◽  
Inverse Limit Spaces ◽  
Positive Dimension ◽  
One Dimensional ◽  
Dimension Group

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

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