Axiom A Research Articles

Combinatorial invariant for Morse–Smale diffeomorphisms on surfaces with orientable heteroclinic

In this paper, we consider a class of orientation-preserving Morse-Smale diffeomorphisms defined on an orientable surface. The papers by Bezdenezhnykh and Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of a heteroclinic intersection. However, such graphs generally do not admit polynomial discriminating algorithms. This article proposes a new approach to the classification of these cascades. For this, each diffeomorphism under consideration is associated with a graph that allows the construction of an effective algorithm for determining whether graphs are isomorphic. We also identified a class of admissible graphs, each isomorphism class of which can be realized by a diffeomorphism of a surface with an orientable heteroclinic. The results obtained are directly related to the realization problem of homotopy classes of homeomorphisms on closed orientable surfaces. In particular, they give an approach to constructing a representative in each homotopy class of homeomorphisms of algebraically finite type according to the Nielsen classification, which is an open problem today.

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

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.

Necessary and sufficient conditions for the conjugacy of Smale regular homeomorphisms

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.

On topological classification of Morse–Smale diffeomorphisms on the sphere S n (n > 3)

We consider the class G(S n ) of orientation preserving Morse–Smale diffeomorphisms of the sphere S n of dimension n > 3, assuming that invariant manifolds of different saddle periodic points have no intersection. For any diffeomorphism f ∈ G(S n ), we define a coloured graph Γ f that describes a mutual arrangement of invariant manifolds of saddle periodic points of the diffeomorphism f. We enrich the graph Γ f by an automorphism P f induced by dynamics of f and define the isomorphism notion between two coloured graphs. The aim of the paper is to show that two diffeomorphisms f, f′ ∈ G(S n ) are topologically conjugated if and only if the graphs Γ f , Γ f ′ are isomorphic. Moreover, we establish the existence of a linear-time algorithm to distinguish coloured graphs of diffeomorphisms from the class G(S n ).

On Realization of Topological Conjugacy Classes of Morse–Smale Cascades on the Sphere $$S^n$$

We consider the class $$G(S^n)$$ of orientation-preserving Morse–Smale diffeomorphisms defined on the sphere $$S^n$$ of dimension $$n\geq 4$$ under the assumption that the invariant manifolds of different saddle periodic points are disjoint. For diffeomorphisms in this class, we describe an algorithm for constructing representatives of all topological conjugacy classes.

Locally connected Smale spaces, pinched spectrum, and infra-nilmanifolds

We show that if (X,f) is a locally connected Smale space (e.g., a basic set of an Axiom-A diffeomorphism) such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then (X,f) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus.

Supporting manifolds for high-dimensional Morse-Smale diffeomorphisms with few saddles

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.

Combinatorial invariant of Morse-Smale diffeomorphisms on surfaces with orientable heteroclinic

In this paper we consider class of orientation-preserving Morse-Smale diffeomorphisms f, given on orientable surface M2. In their articles A.A.~Bezdenezhnich and V. Z. Grines has shown, that such diffeomorfisms contain finite number of heteroclinic orbits. Moreover, the problem of classification for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of heteroclinic intersections. Howewer, these graphs generally do not allow polynomial distinguishing algorithms. In this paper, we propose a new approach to the classification of such cascades. To this end, each considered diffeomorphism f is associated with a graph whose embeddablility in the ambient surface makes it possible to construct an effective algoritm for distinguishing such graphs.

Morse-Smale Surfaced Diffeomorphisms with Orientable Heteroclinic

In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.

Scenario of a Simple Transition from a Structurally Stable 3-Diffeomorphism with a Two-Dimensional Expanding Attractor to a DA Diffeomorphism

The Smale surgery on the three-dimensional torus allows one to obtain a so-called DA diffeomorphism from the Anosov automorphism. The nonwandering set of a DA diffeomor-phism consists of a single two-dimensional expanding attractor and a finite number of source periodic orbits. As shown by V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, the dynamics of an arbitrary structurally stable 3-diffeomorphism with a two-dimensional expanding attrac-tor generalizes the dynamics of a DA diffeomorphism: such a 3-diffeomorphism exists only on the three-dimensional torus, and the two-dimensional attractor is its unique nontrivial basic set, but its nonwandering set may contain isolated saddle periodic orbits together with source periodic orbits. In the present study, we describe a scenario of a simple transition (through elementary bifurcations) from a structurally stable diffeomorphism of the three-dimensional torus with a two-dimensional expanding attractor to a DA diffeomorphism. A key moment in the construction of the arc is the proof that the closure of the separatrices of boundary periodic points of a nontrivial attractor and of isolated saddle periodic points are tamely embedded. This result demonstrates the fundamental difference of the dynamics of such diffeomorphisms from the dynamics of three-dimensional Morse-Smale diffeomorphisms, in which the closure of the separatrices of saddle periodic points may be wildly embedded.

On Embedding of Multidimensional Morse–Smale Diffeomorphisms into Topological Flows

On Embedding of Multidimensional Morse–Smale Diffeomorphisms into Topological Flows

Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

The topological classification of even the simplest Morse–Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possibility of “wild” behavior of separatrices of saddle points. Another difference between Morse–Smale diffeomorphisms in dimension 3 and their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may not be only a point, as in the 2-dimensional case, but also a curve, compact or noncompact. The problem of topological classification of Morse–Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2016 by C. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, and O. Pochinka. The present article is devoted to completing the topological classification of the set MS(M3) of orientation-preserving Morse–Smale diffeomorphisms f on a smooth closed orientable 3-manifold M3. The complete topological invariant for a diffeomorphism f∈MS(M3) is the equivalence class of its scheme Sf which contains information on the periodic data and the topology of embedding of 2-dimensional invariant manifolds of the saddle periodic points of f into the ambient manifold.

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

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.

Periodic expansion in determining minimal sets of Lefschetz periods for Morse\u2013Smale diffeomorphisms

We apply the representation of Lefschetz numbers of iterates in the form of periodic expansion to determine the minimal sets of Lefschetz periods of Morse–Smale diffeomorphisms. Applying this approach we present an algorithmic method of finding the family of minimal sets of Lefschetz periods for N_g, a non-orientable compact surfaces without boundary of genus g. We also partially confirm the conjecture of Llibre and Sirvent (J Diff Equ Appl 19(3):402–417, 2013) proving that there are no algebraic obstacles in realizing any set of odd natural numbers as the minimal set of Lefschetz periods on N_g for any g.

A Combinatorial Invariant of Morse–Smale Diffeomorphisms without Heteroclinic Intersections on the Sphere Sn, n ≥ 4

A Combinatorial Invariant of Morse–Smale Diffeomorphisms without Heteroclinic Intersections on the Sphere Sn, n ≥ 4

A Combinatorial Invariant of Morse-Smale Diffeomorphisms without Heteroclinic Intersections on the Sphere $S^n$, $n\ge 4$

A Combinatorial Invariant of Morse-Smale Diffeomorphisms without Heteroclinic Intersections on the Sphere $S^n$, $n\ge 4$

Conjugacy of Morse–Smale Diffeomorphisms with Three Nonwandering Points

Conjugacy of Morse–Smale Diffeomorphisms with Three Nonwandering Points

Dynamical Systems and Topology of Magnetic Fields in Conducting Medium

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 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.

On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms on products of ℓ-spheres

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the space X=Sℓ×⋯×︸n−timesSℓ, with ℓ>1. Among the quasi-unipotent maps are Morse–Smale diffeomorphisms. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on X; we completely describe this set, for families containing infinitely many Morse–Smale diffeomorphisms. The results of the present article are based on the techniques used in [5], in the computation of the Lefschetz zeta function for quasi-unipotent self maps on the n-dimensional torus.

Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds

We show that, up to topological conjugation, the equivalence class of a Morse–Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.