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О гиперболических аттракторах и репеллерах эндоморфизмов
Нелинейная динамика. 2017. Т. 13. № 4. С. 557-571.
Grines V., Kurenkov E.
Publication based on the results of:
Grines V., Kurenkov E., / Cornell University. Series math "arxiv.org". 2017.
Added: November 13, 2017
Kurenkov E., Журнал Средневолжского математического общества 2017 Т. 19 № 1 С. 60-66
In the article we construct an axiom $A$ endomorphism $f$ of 2-torus with nonwondering set that contains one-dimensional contracting repeller satisfying following properties:
1) $f(\Lambda)= \Lambda$, $f^{-1}(\Lambda)= \Lambda$;
2) $\Lambda$ is locally homeomorphic to the product of the Cantor set and the interval;
3) $T^2\setminus\Lambda$ consist of a countable family of disjoint open disks.
The key idea of construction ...
Added: October 16, 2017
Pochinka O., Левченко Ю. А., Grines V., Нелинейная динамика 2014 Т. 10 № 1 С. 17-33
Consider the class of diffeomorphisms of three-dimensional manifolds and satisfying aksiomA by Smale on the assumption that the non-wandering set of each diffeomorphism consists of surface two-dimensional basic sets. We find interrelations between the dynamics of such a diffeomorphism and the topology of the ambient manifold. Also found that each such diffeomorphism is Ω-conjugate to ...
Added: August 16, 2014
Grines V., Kurenkov E., Журнал Средневолжского математического общества 2018 Т. 20 № 2 С. 159-174
The present paper is devoted to the topological classification of one-dimensional basiс sets of diffeomorphisms satisfying ещ the Smale's axiom A and given on orientable surfaces of negative Euler characteristic equipped with a metric of constant negative curvature. Using Lobachevsky's methods of geometry, each perfect one-dimensional attractor of A-diffeomorphism is uniquely associated with a geodesic ...
Added: June 5, 2018
Grines V., Kurenkov E., Труды Средневолжского математического общества 2016 Т. 18 № 2 С. 16-24
This paper deals with the study of the dynamics in the neighborhood of one-dimensional basic sets of Ck , k ≥ 1 , endomorphism satisfying axiom of A and given on surfaces. It is established that if one-dimensional basic set of endomorphism f has the type (1; 1) and is a onedimensional submanifold without boundary, ...
Added: June 7, 2016
Grines V., Kurenkov E., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2019 Т. 485 № 2 С. 135-138
In the present paper axiom $A$ diffeomorphisms of closed 2-manifolds of genus $p \geq 2$ whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. ...
Added: October 20, 2018
Nozdrinova E., Pochinka O., Tsaplina E., / Cornell University. Series arXiv "math". 2022.
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in ...
Added: December 30, 2022
Pochinka O., Grines V., Zhuzhoma E. V., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2014 Vol. 24 No. 8 P.
In the survey, we consider bifurcations of attracting (or repelling) invariant sets of some classical dynamical systems with a discrete time. ...
Added: September 11, 2014
Grines V., Kurenkov E., Известия РАН. Серия математическая 2020 Т. 84 № 5 С. 40-97
In this paper, we consider orientation-preserving A-diffeomorphisms of orientable surfaces of genus greater than one that contain a one-dimensional, spaciously located perfect attractor. It is shown that the question of the topological classification of the restrictions of diffeomorphisms to such basic sets is reduced to the problem of the topological classification of pseudo-Anosov homeomorphisms with ...
Added: October 30, 2019
Nozdrinova E., Pochinka O., / Cornell University. Серия arXiv "math". 2022.
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in ...
Added: December 30, 2022
Grines V., Kurenkov E., Doklady Mathematics 2019 Vol. 99 No. 2 P. 137-139
Axiom A diffeomorphisms of closed 2-manifold of genus p⩾2 whose nonwandering set contains a perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to a pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of this paper is as follows. Two diffeomorphisms from the given class ...
Added: June 20, 2019
Kazakov A., Bakhanova Y., Козлов А. Д. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2019 Т. 27 № 5 С. 7-52
The main goal of the present paper is an explanation of topical issues of the theory of spiral chaos of three-dimensional flows, i.e. the theory of strange attractors associated with the existence of homoclinic loops to the equilibrium of saddle-focus type, based on the combination of its two fundamental principles, Shilnikov’s theory and universal scenarios ...
Added: October 18, 2019
Grines V., Zhuzhoma E. V., Kurenkov E., Математический сборник 2021 Т. 212 № 5 С. 102-132
It is proved that in each homotopy class of continuous mappings of the two-dimensional torus that induce a hyperbolic action in the fundamental group and do not contain expanding mappings, there exists an A-endomorphism f whose non-wandering set consists of an attracting hyperbolic sink and a nontrivial one-dimensional contracting repeller, which is a one-dimensional orientable ...
Added: April 29, 2021
Volk D., Kleptsyn V., Nonlinearity 2014 Vol. 27 No. 7 P. 1595-1601
In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibres. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the ...
Added: December 22, 2015
Е.В. Жужома, Исаенкова Н. В., В.С. Медведев, Журнал Средневолжского математического общества 2018 Т. 20 № 1 С. 23-29
In the paper we construct some example of smooth dieomorphism of closed manifold. This dieomorphism has one-dimensional (in topological sense) basic set with stable invariant manifold of arbitrary nonzero dimension and the unstable invariant manifold of arbitrary dimension not less than two. The basic set has a saddle type, i.e. is neither attractor nor repeller. ...
Added: May 25, 2018
Nozdrinova E., Журнал Средневолжского математического общества 2017 Т. 19 № 2 С. 91-97
In this paper we consider the class GG orientation preserving gradient-like diffeomorphisms ff defined on a smooth oriented closed surface M2. Establishes that for any such pair of diffeomorphisms there is a dual attractor-repeller Af,Rf, which have a topological dimension not greater than 1 and the orbit space in their Supplement Vf (characteristic space) is ...
Added: June 19, 2017
Romanov A., Известия РАН. Серия математическая 2006 Т. 70 № 5 С. 163-178
<img /> Для эволюционных уравнений параболического типа c гильбертовым фазовым пространством E рассмотрена проблема эффективной (с липшицевой оценкой) конечной параметризации множеств K в E функционалами из E*, или, в иных терминах, проблема линейного липшицева вложения K в конечномерное евклидово пространство. Если K - глобальный аттрактор уравнения, то такого рода параметризация оказывается равносильной конечномерности динамики на K. Получен ряд признаков параметризации (в различных ...
Added: December 6, 2012
Исаенкова Н. В., Zhuzhoma E. V., Труды Математического института им. В.А. Стеклова РАН 2012 Т. 278 № 5 С. 96-101
В статье получена классификация $d$-накрытий степени $d\geq 2$ окружности $S^1$ с точностью до сопряженности с помощью сохраняющих ориентацию гомеоморфизмов. Показано, что полным классификационным инвариантом с точностью до $d$-эквивалентности является наделенное схемой инвариантное счетное множество (отмеченное множество) линейного растягивающего эндоморфизма степени $d$. ...
Added: October 15, 2014
Н.И. Жукова, Mathematical Notes (Rusian Federation) 2013 Т. 93 № 6 С. 994-996
In this paper a unified method for studying foliations with transversal psrsbolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...
Added: September 28, 2014
Barinova M., Гогулина Е. Ю., Pochinka O., Огарёв-Online 2020 № 13 С. 1-10
Smale diagram of Ω-stable diffeomorfism is a graph, which vertices correspond to basic sets and oriented edges sequentially connect vertices of maximal chain. Smale formulated description of acceptable diagrams, which are realized by a diffeomorfism. Present paper proves that an acyclic graph is implemented by an Ω-stable diffeomorfism on surface with nontrivial basic sets. ...
Added: November 23, 2020
On construction of axiom A 3-diffeomorphism with one-dimensional surface attractor-repeller dynamics
Grines V., Pochinka O., Barinova M., Динамические системы 2018 Vol. 8 No. 4 P. 299-305
In this paper we construct an omega-stable diffeomorphism $f$ on closed 3-manifold $M$ so that non-wandering set of $f$ consists of exactly one-dimensional attractor and repeller. All known examples were constructed by Ch. Bonatti, N. Guilman, Sh. Yi. We suggest a new model of the construction of such diffeomorphism. ...
Added: November 21, 2018
Исаенкова Н. В., Zhuzhoma E. V., Челябинский физико-математический журнал 2018 Т. 3 № 3 С. 295-310
We consider a class of Smale — Vietoris A-diffeomorphisms that are defined using basic A-endomorphisms of manifolds, the dimension of which is less than the dimension of the supporting manifolds of A-diffeomorphisms. The class of Smale — Vietoris diffeomorphisms contains DE-mappings of Smale. We show that there is a one-to-one correspondence between the basic sets ...
Added: November 22, 2018
Zhukova N., Математический сборник 2012 Т. 203 № 3 С. 79-106
Доказано, что любое полное конформное слоение (M,F) коразмерности q> 2 является либо римановым, либо (Conf(S^q),S^q)-слоением. Если (M,F) не является римановым слоением, то оно имеет глобальный аттрактор, представляющий собой либо нетривиальное минимальное множество, либо один замкнутый слой или объединение двух замкнутых слоев. При этом компактность многообразия M не предполагается. В частности, каждое собственное полное конформное не риманово ...
Added: September 28, 2014
Grines V., Левченко Ю. А., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2012 Т. 447 № 2 С. 127-129
The paper is devoted to topological classifiication of cascades on 3-manifolds whose nonwandering set consists of surface 2-dimensional basic sets. ...
Added: February 25, 2015