Thanks to the works by R. V. Plykin and V. Z. Grines, the most studied expanding attractors are orientable attractors of codimension one of A-diffeomorphisms of multidimensional closed manifolds and one-dimensional attractors on closed surfaces. In this paper, we prove that there exist closed manifolds of any dimension starting with three, admitting structurally stable diffeomorphisms and diffeomorphisms satisfying Smale’s axiom A, with expanding attractors of arbitrary codimension. For some codimensions the type of manifolds is obtained.

Let M^n, n≥ 3, be a closed orientable n-manifold and 𝔻_k(M^n;a,b,c) the set of axiom A diffeomorp­hisms f: M^n→ M^n satisfying the following conditions: (1) f has k≥ 1 nontrivial basic sets each is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller, and other trivial basic sets which are a sinks, b sources, c saddles; (2) the invariant manifolds of isolated saddles are intersected transversally. We classify the diffeomorphisms from 𝔻_k(M^n;a,b,c) up to the global conjugacy on non-wandering sets for the following subsets 𝕊_k(M^n;a,b,c), ℙ_k(M^n;0,0,1), 𝕄_k(M^n;0,0,1) of 𝔻_k(M^n;a,b,c) where 𝕊_k(M^n;a,b,c) satisfies to the following conditions: (1_𝕊) every nontrivial basic set of any f∈𝕊_k(M^n;a,b,c) is uniquely bunched, and there is at least one nontrivial attractor and at least one nontrivial repeller, i.e. k≥ 2; (2_𝕊) c≥ 1 and all isolated saddles have the same Morse index belonging to {1,n-1}. The subset ℙ_k(M^n;0,0,1)⊂𝔻_k(M^n;0,0,1) satisfies to the following conditions: (1_ℙ) any boundary point of f∈ℙ_k(M^n;0,0,1) is fixed; (2_ℙ) a unique isolated saddle has Morse index different from {1,n-1}. The subset 𝕄_k(M^n;0,0,1)⊂𝔻_k(M^n;0,0,1) satisfies to the following conditions: (1_𝕄) any boundary point of f∈𝕄_k(M^n;0,0,1) is fixed; (2_𝕄) a unique isolated saddle has Morse index belonging to {1,n-1}. The classification is based on a description of topological structure of supporting manifolds M^n.

We prove that an n -sphere 𝕊^n , n⩾ 2 , admits structurally stable diffeomorphisms 𝕊^n→𝕊^n with nonorientable expanding attractors of any topological dimension d∈{1,…,[n/2]} where [x] is the integer part of x . In addition, any n -sphere 𝕊^n , n⩾ 3 , admits axiom A diffeomorphisms 𝕊^n→𝕊^n with orientable expanding attractors of any topological dimension d∈{1,…,[n/3]} . We prove that an n -torus 𝕋^n , n⩾ 2 , admits structurally stable diffeomorphisms 𝕋^n→𝕋^n with orientable expanding attractors of any topological dimension d∈{1,…,n-1} . We also prove that, given any closed n -manifold M^n , n⩾ 2 , and any d∈{1,…,[n/2]} , there is an axiom A diffeomorphism f:M^n→ M^n with a d -dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.

We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^{n}$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].

We introduce high-dimensional Morse-Smale systems with king-saddles. Every saddle of such system has a separatrix with heteroclinic intersections. We get the necessary and sufficient conditions of conjugacy of Morse-Smale diffeomorphisms with king-saddles. For every polar Morse-Smale system with two saddles on the n-sphere Sn, one proves the existence of a king-saddle. This allows to get the necessary and sufficient conditions of conjugacy for such Morse-Smale diffeomorphisms on Sn, n≥4. We give a simple sufficient condition of the existence of heteroclinic intersections for Morse-Smale systems on S3.

Anatolii Mikhailovich Stepin, a prominent scientist and educator, expert in dynamical systems and ergodic theory, passed away on 7 November 2020. His death was a bitter loss for his family, students, colleagues, and mathematicians at large. He was born in Moscow on 20 July 1940. During the war he was evacuated to Chelyabinsk together with his mother, while his father, a chemical engineer, worked at a defense factory. After the war their family returned to Moscow. Anatolii graduated from Moscow School no. 434 with a gold medal and, intending to follow in his father’s footsteps, enrolled at the Moscow Power Engineering Institute, the Division of Thermal Physics. At that time M. I. Vishik, who was well known for his work on partial differential equations, read mathematics courses at the Institute. The talented student became interested in these lectures, and after his 3rd year at Power Engineering Institute, he decided to transfer to the Faculty of Mechanics and Mathematics at Moscow State University. After solving the tricky problem of being transferred to another university, Stepin became a student at the Department of Theory of Functions and Functional Analysis, and chose F.A. Berezin to be his scientific advisor. In addition to Berezin’s research seminar, Stepin was also

В статье определяется класс диффеоморфизмов Морса-Смейла с доминантным седлом и приводятся необходимые и достаточные условия сопряженности таких диффеоморфизмов. Показывается, что полярные диффеоморфизмы Морса-Смейла $n$-мерной сферы $\mathbb S^n$, $n\ge 4$, неблуждающее множество которых состоит из четырех точек, имеют доминантные седла. В качестве следствия получаем необходимые и достаточные условия сопряженности таких диффеоморфизмов. Мы приводим примеры полярных диффеоморфизмов $\mathbb S^n\to\mathbb S^n$ Морса-Смейла с указанным неблуждающим множеством. Библиография: 21 название.

Let f t be a flow satisfying Smale’s Axiom A (in short, A-flow) on a closed orientable three-manifold M 3, and Ω a two-dimensional basic set of f t . First, we prove that Ω is either an expanding attractor or contracting repeller. Next, one considers an A-flow f t with a two-dimensional non-mixing attractor Λ a . We construct a casing M(Λ a ) of Λ a that is a special compactification of the basin of Λ a by a collection of circles L(Λ a ) = {l 1, …, l k } such that M(Λ a ) is a closed three-manifold and L(Λ a ) is a fibre link in M(Λ a ). In addition, f t is extended on M(Λ a ) to a nonsingular structurally stable flow with the non-wandering set consisting of the attractor Λ a and the repelling periodic trajectories l 1, …, l k . We show that if a closed orientable three-manifold M 3 has a fibred link L = {l 1, …, l k } then M 3 admits an A-flow f t with the non-wandering set containing a two-dimensional non-mixing attractor and the repelling isolated periodic trajectories l 1, …, l k . This allows us to prove that any closed orientable n-manifold, n ⩾ 3, admits an A-flow with a two-dimensional attractor. We prove that the pair M(Λa);L(Λa) consisting of the casing M(Λ a ) and the corresponding fibre link L(Λ a ) is an invariant of conjugacy of the restriction ft|Ws(Λa) of the flow f t on the basin of the attractor Λ a .

We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse–Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point–charge magnetic field model, we consider the problem of the separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

The paper describes the topological structure of closed manifolds of dimension $$\ge4$$ that admit Morse–Smale diffeomorphisms whose nonwandering sets contain arbitrarily many sink periodic points, arbitrarily many source periodic points, and two saddle periodic points. The underlying manifolds of Morse–Smale diffeomorphisms with fewer saddle periodic points are also described.

The class of Smale regular homeomorphisms of closed topological manifolds, with nonwandering set consisting of a finite number of periodic orbits of hyperbolic type, is considered. This class contains the Morse-Smale diffeomorphisms of smooth closed manifolds. For two Smale regular homomorphisms necessary and sufficient conditions for being conjugate are presented. Bibliography: 26 titles.

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 $\mathbb{S}^{2}$.

Доказывается, что в каждом гомотопическом классе непрерывных отображений двумерного тора, индуцирующих гиперболическое действие в фундаментальной группе и не содержащих растягивающих отображений, существует $A$-эндоморфизм $f$, неблуждающее множество которого состоит из притягивающего гиперболического стока и нетривиального одномерного сжимающегося репеллера, который является одномерной ориентируемой ламинацией, локально гомеоморфной прямому произведению канторова множества на отрезок. Более того, неустойчивое $Df$-инвариантное подрасслоение касательного пространства к репеллеру обладает свойством единственности. Библиография: 23 названия.

We prove that given any closed 3-manifold M-3, there is an A-flow f(t) on M-3 such that the non-wandering set NW (f(t)) consists of 2-dimensional non-orientable expanding attractor and trivial basic sets.

We prove that given any closed $n$-manifold $M^n$, $n\geq 4$, there is an A-flow $f^t$ on $M^n$ such that the non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor (the both, orientable and non-orientable) and trivial basic sets. For 3-manifolds, we prove that given any closed 3-manifold $M^3$, there is an A-flow $f^t$ on $M^3$ such that the non-wandering set $NW(f^t)$ consists of a non-orientable 2-dimensional expanding attractor and trivial basic sets. Moreover, there is a nonsingular A-flow $f^t$ on a 3-sphere such that the non-wandering set $NW(f^t)$ consists of an orientable 2-dimensional expanding attractor and trivial basic sets (isolated periodic trajectories).

We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.

We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension 4 and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly μ node and ν saddle equilibrium points of indices equal to either 1 or n — 1. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius $$g=\frac{\nu-\mu+2}{2}$$ . We also suggest an algorithm for constructing gradientlike flows on closed manifolds of dimension 3 and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices.

Ìíîãîìåðíûå ñîëåíîèäàëüíûå èíâàðèàíòíûå ìíîaeåñòâà ñåäëîâîãî òèïà c ⃝ Å. Â. AEóaeîìà , Í. Â. Èñàåíêîâà , Â. Ñ. Ìåäâåäåâ !Àííîòàöèÿ. ñòàòüå ìû ñòðîèì ïðèìåð ãëàäêîãî äèôôåîìîðôèçìà çàìêíóòîãî ìíîãîîáðàçèÿ, êîòîðûé èìååò îäíîìåðíîå (â òîïîëîãè÷åñêîì ñìûñëå) ñîëåíîèäàëüíîå áàçèñíîå ìíîaeåñòâî ñ óñòîé÷èâûì èíâàðèàíòíûì ìíîãîîáðàçèåì ïðîèçâîëüíîé íåíóëåâîé (íàïåðåä çàäàííîé) ðàçìåðíîñòè è óñòîé÷èâûì èíâàðèàíòíûì ìíîãîîáðàçèåì ïðîèçâîëüíîé ðàçìåðíîñòè, áîëüøåé èëè ðàâíîé äâóì