scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

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.

scholarly journals On 4-dimensional flows with wildly embedded invariant manifolds of a periodic orbit

2020 ◽  
Vol 5 (2) ◽  
pp. 261-266
Author(s):  
Olga V. Pochinka ◽  
Danila D. Shubin
Keyword(s):  
Periodic Orbit ◽  
Unstable Manifold ◽  
Invariant Manifolds ◽  
Topological Classification

AbstractIn the present paper we construct an example of 4-dimensional flows on 𝕊3 × 𝕊1 whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

scholarly journals Topological Moduli Space for Germs of Holomorphic Foliations

2018 ◽  
Vol 2020 (23) ◽  
pp. 9228-9292
Author(s):  
David Marín ◽  
Jean-François Mattei ◽  
Éliane Salem
Keyword(s):  
Moduli Space ◽  
Topological Structure ◽  
Finite Dimension ◽  
Functional Space ◽  
Topological Classification ◽  
Topological Invariants ◽  
Holomorphic Foliations ◽  
Infinite Dimensional ◽  
Singular Foliations

Abstract This work deals with the topological classification of germs of singular foliations on $(\mathbb{C}^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho–Sad indices, and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite-dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological structure.

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.

RHYNCHOLITES AND THE PROBLEM OF NARROW AND BROAD CONCEPTION OF TAXONS

2018 ◽  
pp. 12-17 ◽  
Author(s):  
I. R. Khuzina ◽  
V. N. Komarov
Keyword(s):  
Sexual Dimorphism ◽  
Morphological Characteristics ◽  
Mass Scale ◽  
Individual Variability ◽  
Point Of View ◽  
The Other ◽  
New Taxon ◽  
Artificial System ◽  
Broad Understanding

The paper considers a point of view, based on the conception of the broad understanding of taxons. According to this point of view, rhyncholites of the subgenus Dentatobeccus and Microbeccus are accepted to be synonymous with the genus Rhynchoteuthis, and subgenus Romanovichella is considered to be synonymous with the genus Palaeoteuthis. The criteria, exercising influence on the different approaches to the classification of rhyncholites, have been analyzed (such as age and individual variability, sexual dimorphism, pathological and teratological features, degree of disintegration of material), underestimation of which can lead to inaccuracy. Divestment of the subgenuses Dentatobeccus, Microbeccus and Romanovichella, possessing very bright morphological characteristics, to have an independent status and denomination to their synonyms, has been noted to be unjustified. An artificial system (any suggested variant) with all its minuses is a single probable system for rhyncholites. The main criteria, minimizing its negative sides and proving the separation of the new taxon, is an available mass-scale material. The narrow understanding of the genus, used in sensible limits, has been underlined to simplify the problem of the passing the view about the genus to the other investigators and recognition of rhyncholites for the practical tasks.

LEGAL, MORAL AND ETHICAL SIDE OF THE OFFENSE, DISCREDITING HONOR EMPLOYEE THE PENAL SYSTEM

2020 ◽  
Vol 10 (2) ◽  
pp. 213-218
Author(s):  
OKSANA KOCHKINA ◽  
◽  
OLGA MARCHUK ◽  
Keyword(s):  
Point Of View ◽  
Professional Activity ◽  
Penal System ◽  
Moral Norms ◽  
Ethical Aspects ◽  
Self Assessment ◽  
Personal Qualities ◽  
Executive System ◽  
Correction System

The article examines the legal and moral and ethical aspects of a misdemeanor that discredits the honor of an employee of the criminal Executive system. The considered reason for dismissal has the main feature associated with the integration of legal and moral norms, which often raises a lot of questions about the attribution of a particular offense to this basis. Using the analysis of normative legal acts, the authors attempt to identify the signs that contribute to the separation of the studied grounds for dismissal from all their diversity. The classification of offenses that discredit the honor of an employee of the criminal Executive system is presented, which allows to systematize and organize the knowledge obtained about the considered grounds for dismissal. The analysis of a misdemeanor that defames the honor of an employee of the penal system from a moral and ethical position gives an understanding, first of all, that it does not have a clear regulation from the point of view of the law, but the consequences of committing such a misdemeanor are clearly legal. The concepts of “honor” and “dignity” are considered as ethical categories and are analyzed as personal qualities that are manifested in an employee of the penal correction system during the period of service. These categories in the behavior of a person or employee are manifested both externally (assessment from the outside) and internally (self-assessment). The article describes the value orientation of an employee of the criminal Executive system to ethical standards in professional activity, which is an integral part of the moral and ethical side of a misdemeanor that discredits the honor of an employee.

REFLECTION OF THE CURRENT STATE OF PUBLICISTIC JOURNALISM GENRES IN THE SLOVAK MEDIA ENVIRONMENT

2020 ◽  
Author(s):  
Zuzana Kvetanová
Keyword(s):  
Discourse Analysis ◽  
Point Of View ◽  
Media Theory ◽  
Logical Analysis ◽  
Periodical Press ◽  
Media Environment ◽  
Current State ◽  
Journalistic Practice ◽  
Opinion Journalism

The submitted study addresses the topic of the current state of the opinion journalism and its genres in the Slovak periodical press. The author draws attention to the question of classification of the opinion journalism of a rational and emotional type from the genre categorization point of view and, simultaneously, reflects on its application in the present journalistic practice. This brings a certain rate of confrontation between the defined theoretical premises and their subsequent practical (non-)implementation. The main objective of the study is to clarify the presence of genres of analytical and literary opinion journalism stated by media theory in the environment of the Slovak periodicals. Presentation of the basic terminological axis and the related explication of journalism genres included in the opinion journalism constitute the secondary objectives of the paper. For the purposes of achieving the set objectives, the author uses methods of logical analysis of text in combination with discourse analysis. Similarly, she predicts the evident presence of the phenomenon of hybridization in the Slovak journalistic practice.

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