Axiom A Research Articles

Nonexistence of energy function for surface diffeomorphisms with horseshoe basic sets

The Lyapunov function is a powerful tool for studying dynamical systems. In particular, the existence of such a function for any dynamical system given on compact manifold, established by C. Conley, is the basis for proving the Ω-stability criterion of dynamical systems. Moreover, the Lyapunov function, whose set of critical points of aligns with the chain-recurrent set of the system, which called an energy function, qualitatively determines the dynamics of the system. For continuous-time systems, which models dissipative processes, such a function naturally arises as an energy of the system. The existence of an energy function for arbitrary flows on compact manifolds is known. Examples of Morse–Smale diffeomorphism without energy function exist on n-dimensional manifolds starting with n>2. For systems with chaotic dynamics on surfaces, the existence of energy functions has also been established in cases where the dimension of non-trivial basic sets is greater than or equal to 1. When the chain-recurrent set of a diffeomorphism contains at least one zero-dimensional non-trivial basic set, the question of the existence of the energy function is still open. To date, it is known that the answer is negative if the zero-dimensional non-trivial basic set does not contain pairs of conjugated points. The present paper is devoted to the question of existence of an energy function for systems having a basic set with pairs of conjugated points. The primary outcome of this study is that there is no energy function for a dynamical system containing a Smale horseshoe as a basic set.

On Morse–Smale diffeomorphisms on simply connected manifolds

We study relations between a structure of non-wandering set of a Morse–Smale diffeomorphism f and its carrying closed manifold Mn. We prove that if f has no any saddle periodic points with one-dimensional unstable manifolds, and for any periodic point σ of Morse index (n−1) its unstable manifolds do not intersect stable invariant manifolds of saddle periodic points different from σ, then Mn is simply connected. This fact does not follow from Morse inequalities that give only restrictions on homology groups of Mn.

Characteristic Space of Orbits of Morse–Smale Diffeomorphisms on Surfaces

Characteristic Space of Orbits of Morse–Smale Diffeomorphisms on Surfaces

Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the "source-sink" form, that is, by singling out a dual attractor-repeller pair, consisting of the attracting and repelling sets for all other trajectories of the system. A choice of an attractor-repeller dual pair so that the space of orbits in their complement (the characteristic space of orbits) is connected paves the way for finding complete topological invariants of the dynamical system. In this way, in particular, several classification results for Morse-Smale systems were obtained. Thus, a complete topological classification of Morse-Smale 3-diffeomorphisms is essentially based on the existence of a connected characteristic space of orbits associated with the choice of a one-dimensional dual attractor-repeller pair. For Morse-Smale diffeomorphisms with heteroclinic points on surfaces, there are examples in which the characteristic spaces of orbits are disconnected in all cases. In this paper, we prove a criterion for the existence of a connected characteristic space of orbits for gradient-like (without heteroclinic points) diffeomorphisms on surfaces. This result implies, in particular, that any orientation-preserving diffeomorphism admits a connected characteristic space. For an orientable surface of any kind, we also construct an orientation-changing gradient-like diffeomorphism that does not have a connected characteristic space. On any non-orientable surface of any kind, we also construct a gradient-like diffeomorphism which does not admit a connected characteristic space.

Bicolor Graph of Morse-Smale Cascades on Manifolds of Dimension Three

The purpose of this study is to single out a class of Morse-Smale cascades (diffeomorphisms) with a three-dimensional phase space that allow topological classification using combinatorial invariants. In the general case, an obstacle to such a classification is the possibility of wild embedding of separatrix closures in the ambient manifold, which leads to a countable set of topologically nonequivalent systems. To solve the problem, we study the orbit space of a cascade. The ambient manifold of a diffeomorphism can be represented as a union of three pairwise disjoint sets: a connected attractor and a repeller whose dimension does not exceed one, and their complement consisting of wandering points of a cascade called the characteristic set. It is known that the topology of the orbit space of the restriction of the Morse-Smale diffeomorphism to the characteristic set and the embedding of the projections of two-dimensional separatrices into it is a complete topological invariant for Morse-Smale cascades on three-dimensional manifolds. Moreover, a criterion for the inclusion of Morse-Smale cascades in the topological flow was obtained earlier.These results are used in this paper to show that the topological conjugacy classes of Morse-Smale cascades that are included in a topological flow and do not have heteroclinic curves admit a combinatorial description. More exactly, the class of Morse-Smale diffeomorphisms without heteroclinic intersections, defined on closed three-dimensional manifolds included in topological flows and not having heteroclinic curves, is considered. Each cascade from this class is associated with a two-color graph describing the mutual arrangement of two-dimensional separatrices of saddle periodic points. It is proved that the existence of an isomorphism of two-color graphs that preserves the color of edges is a necessary and sufficient condition for the topological conjugacy of cascades. It is shown that the speed of the algorithm that distinguishes two-color graphs depends polynomially on the number of its vertices. An algorithm for constructing a representative of each topological conjugacy class is described.

Structural stability and rate of convergence of global attractors

In this paper, we prove that a Lipschitz perturbed map of a gradient Morse–Smale diffeomorphism T on Rm has the Lipschitz shadowing property on a neighborhood of the global attractor AT, and apply the result to get the structural stability and the rate of convergence of global attractors of Chafee–Infante equations under Lipschitz perturbations of the domain and equation.

Determination of the Homotopy Type of a Morse-Smale Diffeomorphism on an Orientable Surface by a Heteroclinic Intersection

Determination of the Homotopy Type of a Morse-Smale Diffeomorphism on an Orientable Surface by a Heteroclinic Intersection

Link as a complete invariant of Morse-Smale 3-diffeomorphisms

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S3 . For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles embedded in some closed 3-manifold whose fundamental group admits an epimorphism to the group Z . Such an invariant is called the scheme of the gradient-like diffeomorphism f:S3→S3 . We single out a class G of diffeomorphisms whose complete invariant is a topologically simpler object, namely, the link of essential knots in the manifold S2×S1 . The diffeomorphisms under consideration are determined by the fact that their non-wandering set contains a unique source, and the closures of stable saddle point manifolds bound three-dimensional balls with pairwise disjoint interiors. We prove that, in addition to the closure of these balls, a diffeomorphism of the class G contains exactly one nonwandering point, which is a fixed sink. It is established that the total invariant of topological conjugacy of class G diffeomorphisms is the space of orbits of unstable saddle separatrices in the basin of this sink. It is shown that the space of orbits is a link of non-contractible knots in the manifold S2×S1 and that the equivalence of links is tantamount to the equivalence of schemes. We also provide a realization of diffeomorphisms of the considered class along an arbitrary link consisting of essential nodes in the manifold S2×S1 .

Shadowing, topological entropy and recurrence of induced Morse–Smale diffeomorphism

Shadowing, topological entropy and recurrence of induced Morse–Smale diffeomorphism

Elementary proof for the existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

Using only elementary techniques we prove that for any $ C^r $ diffeomorphism, $ f $, of a compact manifold of dimension $ d>2 $, $ 1\leq r\leq \infty $, admitting a transverse homoclinic intersection, we can find a $ C^1 $-open neighborhood of $ f $ containing a $ C^1 $-open and $ C^r $-dense set of $ C^r $ diffeomorphisms which have a periodic point with real and simple spectrum. We use this fact to prove that $ C^r $-generically among $ C^r $ diffeomorphisms with horseshoes, we have density of periodic points with real and simple spectrum inside the horseshoe. As a corollary, we obtain that generically in the $ C^1 $-topology the unique obstruction to the existence of periodic points with real and simple spectrum are the Morse-Smale diffeomorphisms with all the periodic points admitting non-real eigenvalues.

Classification of Morse–Smale Diffeomorphisms with a Finite Set of Heteroclinic Orbits on Surfaces

Classification of Morse–Smale Diffeomorphisms with a Finite Set of Heteroclinic Orbits on Surfaces

Polynomial entropy of Morse-Smale diffeomorphisms on surfaces

A classical problem in dynamical systems is to measure the complexity of a map in terms of its orbits, and one of the main concepts used to achieve this goal is entropy. Nonetheless, many interesting families of dynamical systems have every element with zero-entropy. One of these are Morse-Smale diffeomorphisms. In this work, we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces, based on which we deduce their polynomial entropy. We also apply our technique to compute the dispersion of the orbits of maps on the border of chaos with mild dissipation.

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 be embedded in topological flows. The progress achieved in solving of Palis’s problem in dimension three is associated with the recently obtained complete topological classification of Morse–Smale diffeomorphisms on threedimensional 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.

High-dimensional Morse-Smale systems with king-saddles

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.

Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection

Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse–Smale Diffeomorphisms with Orientable Heteroclinic Intersection

Realization of Homeomorphisms of Surfaces of Algebraically Finite Order by Morse-Smale Diffeomorphisms with Orientable Heteroclinic Intersection

Согласно классификации Тeрстона множество гомотопических классов гомеоморфизмов замкнутых ориентируемых поверхностей отрицательной кривизны разбивается на четыре непересекающихся подмножества $T_1$, $T_2$, $T_3$, $T_4$. Гомотопический класс из каждого подмножества характеризуется существованием в нем гомеоморфизма (канонической формы Тeрстона), имеющего в точности один из следующих типов соответственно: периодический гомеоморфизм, приводимый непериодический гомеоморфизм алгебраически конечного порядка, приводимый гомеоморфизм, не являющийся гомеоморфизмом алгебраически конечного порядка, псевдоаносовский гомеоморфизм. Канонические формы Тeрстона не являются структурно устойчивыми диффеоморфизмами. Поэтому естественно возникает задача построения простейших (в определенном смысле) структурно устойчивых диффеоморфизмов в каждом гомотопическом классе. А.Н. Безденежных и В.З. Гринес построили градиентно-подобный диффеоморфизм в каждом гомотопическом классе из $T_1$. А.Ю. Жиров и Р.В. Плыкин анонсировали метод построения структурно устойчивого диффеоморфизма в каждом гомотопическом классе из $T_4$. Неблуждающее множество этого диффеоморфизма состоит из конечного числа источниковых орбит и единственного одномерного аттрактора. В настоящей работе описано построение структурно устойчивого диффеоморфизма в каждом гомотопическом классе из $T_2$. Построенный представитель является диффеоморфизмом Морса-Смейла с ориентируемым гетероклиническим пересечением.

Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms

Abstract. According to Thurston’s classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence of a homeomorphism called Thurston’s canonical form, namely: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of an algebraically finite order, and a pseudo-Anosov homeomorphism. Thurston’s canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In this paper, the problem posed is solved for torus homeomorphisms. In each homotopy class, structurally stable representatives are analytically constructed, namely, a gradient-like diffeomorphism, a Morse-Smale diffeomorphism with an orientable heteroclinic, and an Anosov diffeomorphism, which is a particular case of a pseudo-Anosov diffeomorphism.

Underlying Manifolds of High-Dimensional Morse–Smale Diffeomorphisms with Two Saddle Periodic Points

Underlying Manifolds of High-Dimensional Morse–Smale Diffeomorphisms with Two Saddle Periodic 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 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.