Smale Diffeomorphisms Research Articles

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.

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

Conjugacy of Morse–Smale Diffeomorphisms with Three Nonwandering Points

Conjugacy of Morse–Smale Diffeomorphisms with Three Nonwandering Points

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.

Kupka–Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

We construct an explicit example of a family of non-uniformly hyperbolic diffeomorphisms, at the boundary of a set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a Cantor set of choices of the second one. For a non-countable subset of these choices, the second leaf is not periodic and the diffeomorphism is Kupka–Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. The bifurcating system is Hölder-conjugated to a subshift of finite type; thus every Hölder potential admits a unique equilibrium state associated with it.

Realization of Morse–Smale diffeomorphisms on 3-manifolds

The paper presents a realization of an orientation-preserving Morse–Smale 3-diffeomorphism in each class of the topological conjugacy by means of an abstract scheme.

On embedding Morse–Smale diffeomorphisms on the sphere in topological flows

On embedding Morse–Smale diffeomorphisms on the sphere in topological flows

On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. II: The general case

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n-dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n-dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7]. The methods used are based on the arithmetical properties of the number n.

On Diffeomorphisms of Compact 2-Manifolds with All Nonwandering Points Being Periodic

The aim of the present paper is to study conditions under which all the nonwandering points are periodic points, for a discrete dynamical system of two variables defined on a compact manifold. We include a survey of known results in all dimensions, and study the remaining open question in dimension two. We present two results, one negative and one positive. The negative result: we construct a Kupka–Smale diffeomorphism in [Formula: see text] (which can be extended to a diffeomorphism of the sphere) with a closed set of periodic points that differs from the set of nonwandering points. The positive result: we present a condition on the widely studied Hénon family which guarantees that all nonwandering points are periodic. Finally, we close by describing what future work may be needed to resolve our broad goals.

Topological Classification of Morse–Smale Diffeomorphisms Without Heteroclinic Intersections

We study the class G(M n ) of orientation-preserving Morse–Smale diffeomorfisms on a connected closed smooth manifold M n of dimension n ≥ 4 which is defined by the following condition: for any f ∊ G(M n ) the invariant manifolds of saddle periodic points have dimension 1 and (n − 1) and contain no heteroclinic intersections. For diffeomorfisms in G(M n ) we establish the topoligical type of the supporting manifold which is determined by the relation between the numbers of saddle and node periodic orbits and obtain necessary and sufficient conditions for topological conjugacy. Bibliography: 14 titles.

On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus

We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the th cyclotomic polynomial, when is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for quasi-unipotent maps on , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse–Smale diffeomorphisms of .

On the set of periods for the Morse–Smale diffeomorphisms on the disc with N holes

Let be the two-dimensional disc with n holes. We assume that is compact. For every homological class of Morse–Smale diffeomorphisms on without periodic points in its boundary, we provide an algorithm for characterizing its minimal set of Lefschetz periods. We give the complete classification of these sets of periods for . The main tool used for this characterization is the Lefschetz zeta function.

Some consequences of the shadowing property in low dimensions

AbstractWe consider low-dimensional systems with the shadowing property and we study the problem of existence of periodic orbits. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $\epsilon $-transitive class, and in contrast we provide an example of a ${C}^{\infty } $ Kupka–Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally, we consider the case of transitive endomorphisms of the circle, and we prove that the $\alpha $-Hölder shadowing property with $\alpha \gt 1/ 2$ implies that the system is conjugate to an expanding map.

Minimal sets of periods for Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary

We study the minimal set of (Lefschetz) periods of the C 1 Morse–Smale diffeomorphisms on a non-orientable compact surface without boundary inside its class of homology. In fact our study extends to the C 1 diffeomorphisms on these surfaces having finitely many periodic orbits, all of them hyperbolic and with the same action on the homology as the Morse–Smale diffeomorphisms. We mainly have two kinds of results. First, we completely characterize the possible minimal sets of periods for the C 1 Morse–Smale diffeomorphisms on non-orientable compact surface without boundary of genus g with . But the proof of these results provides an algorithm for characterizing the possible minimal sets of periods for the C 1 Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary of arbitrary genus. Second, we study what kind of subsets of positive integers can be minimal sets of periods of the C 1 Morse–Smale diffeomorphisms on a non-orientable compact surface without boundary.

Morse–Smale systems with few non-wandering points

Let MSflow(Mn,k) and MSdiff(Mn,k) be Morse–Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold Mn. For k=3, we show that the only values of n possible are n∈{2,4,8,16}, and M2 is the projective plane. For n⩾4, Mn is simply connected and orientable. We prove that the closure of any separatrix of ft∈MSflow(Mn,3) is a locally flat n2-sphere while there is ft∈MSflow(Mn,4) such that the closure of separatrix of ft is a wildly embedded codimension two sphere. This allows us to classify flows from MSflow(M4,3). For n⩾6, one proves that the closure of any separatrix of f∈MSdiff(Mn,3) is a locally flat n2-sphere while there is f∈MSdiff(M4,3) such that the closure of any separatrix is a wildly embedded 2-sphere.

Embedding in a flow of Morse-Smale diffeomorphisms on manifolds of dimension higher than two

We say that a diffeomorphism f : Mn → Mn embeds in a topological (smooth) flow if there exists a topological (smooth) flow Xt on Mn such that f is the time-one map of Xt. It follows from results of the papers [1] and [2], in which the structural stability of Morse–Smale diffeomorphisms was proved, that, for any manifold Mn, there exists an open (in Diff(Mn)) set of Morse–Smale diffeomorphisms embeddable in a topological flow. In [1], Palis proved the following necessary condition for a Morse– Smale diffeomorphism f to embed in a topological flow:

Cyclotomic polynomials and minimal sets of Lefschetz periods

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the 's are positive integers, 's are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855).