Dynamical Systems on 2- and 3-Manifolds

In this chapter we present without proofs notions and facts on the dynamical systems which are necessary to understand this book. We recall the notion of an invariant set and show the most important examples of such sets: fixed and periodic points, \(\omega -\) and \(\alpha -\)limit sets, wandering and nonwandering sets, chain recurrent sets, topologically transitive sets. We discuss the notion of stability of a dynamical system with respect to one of its characteristics, structural stability and \(\varOmega -\)stability in particular. We consider hyperbolic invariant sets, recall the theorem on existence of the stable and the unstable manifold for a point of such a set. We recall Hartman-Grobman theorem that a diffeomorphism in a neighborhood of a hyperbolic periodic point is topologically conjugate to its linearizion. We give the topological classification of hyperbolic fixed points. We present a brief explanation of the results of the “epoch of the hyperbolic revolution” begun in 1960s with the classical works by S. Smale and D. Anosov. We show the relations between the nonwandering set, the chain recurrent set and the limit set of a dynamical system (diffeomorphism). We present Smale’s spectral decomposition theorem which allows us to represent a hyperbolic nonwandering set of a diffeomorphism as the union of the disjoint closed transitive (basic) sets if the nonwandering set is the closure of the periodic points. We recall the criteria of structural and \(\varOmega -\)stability. The concepts of the symbolic dynamics, the reverse limit and the solenoid are presented as the means to describe the restriction of a dynamical system to its invariant set with complex dynamics. For more details see for example the books [17, 24, 28, 33, 43, 46], the surveys [3‐6, 48] and the papers [1, 7‐9, 11, 13, 14, 18, 23, 26, 27, 29, 30, 34, 47, 51].

In this chapter we consider an important class of discrete structurally stable systems which adequately describe processes with regular dynamics, Morse-Smale diffeomorphisms. We present with proofs the properties of Morse-Smale diffeomorphisms which are necessary for the topological classification. The asymptotic behavior and the embedding into the ambient manifold (the phase space) of the stable and the unstable manifolds of the saddle periodic points plays the key role in understanding of the dynamics of such diffeomorphisms. To describe the topological invariants which reflect these properties we consider the space of wandering orbits which belong to some specially chosen invariant sets of the diffeomorphism. We describe the important (for the subsequent results) construction of the sequence of the “attractor-repeller” pairs suggested by C. Conley. This construction is based on introduction of an order on the set of the periodic orbits which satisfies the Smale partial relation. The proof of existence of a trapping neighborhood of an attractor (a repeller) relies on the local Morse-Lyapunov function constructed in this chapter. All the proofs are presented for the class \(MS(M^n)\) of the orientation preserving Morse-Smale diffeomorphisms \(f:M^n\rightarrow M^n\) on an orientable manifold \(M^n\). The results are partly announced and proved in the surveys [1‐3, 9] and the papers [4‐8].

In this chapter we study the periodic data (the period, the Morse index, the orientation type) of the periodic orbits of gradient-like diffeomorphisms on orientable surfaces. Though such diffeomorphisms are similar in many ways to structurally stable flows on surfaces they have one property which makes them considerably different. This property is a possible non-trivial periodic action of the diffeomorphism in the fundamental group of the surface. The study of admissible collections of periodic data made it possible to solve the problem of realization of gradient-like diffeomorphisms. It also showed the interrelation between the dynamics of such diffeomorphisms and periodic transformations of surfaces whose classification is an important part of Nielsen-Thurston theory. In the present chapter we introduce a topological invariant for gradient-like diffeomorphisms on orientable surfaces. This invariant is a graph similar to that of Peixoto for structurally stable flows without cycles. We prove that such a graph equipped with a permutation of the set of the vertices completely determines the class of topological conjugacy of a gradient-like diffeomorphism on a surface. Moreover, we construct another complete topological invariant for these diffeomorphisms (a scheme) which is based on the representation of the dynamics of a diffeomorphism as “attractor-repeller” and on the subsequent study of the space of wandering orbits. We show that the class of topological conjugacy of a gradient-like diffeomorphism is determined (up to a conjugating homeomorphism) by a collection of 2-tori each of which has a family of circles embedded into it. The results on the topological classification of special classes of the Morse-Smale diffeomorphisms on 2-manifolds can be found in [1‐6, 8, 9].

The Morse-Smale diffeomorphisms on 3-manifolds are in marked contrast to the Morse-Smale flows and diffeomorphisms on 2-manifolds because of the possibility of wild embedding of the separatrices of the saddle points. In this chapter we prove the criteria of the tame embedding of 1- and 2-dimensional separatrices. We give the complete study of the simplest Morse-Smale diffeomorphisms with the wandering set of four points if exactly one of them is the saddle (the Pixton diffeomorphisms). We show that the invariant manifolds of the saddle point of such diffeomorphisms can be wildly embedded. It implies that there are countably many topologically non-conjugate simplest diffeomorphisms. The topological invariant in this case is the knot in the characteristic space homeomorphic to \(\mathbb S^2\times \mathbb S^1\). Then we study the bifurcations through which the transition from one class of topologically conjugate diffeomorphisms to another occurs. The distinctive specialty of the new bifurcation is that the structure on the non-wandering set does not change but the qualitative change of the diffeomorphism is due to the change of the type of the embedding of the separatrices of the saddle points. This problem is connected to the problem of J. Palis and C. Pugh [8], that is to find a smooth curve with some “good” properties (finitely many bifurcations for instance) that joins two structurally stable systems (flows or diffeomorphisms). S. Newhouse and M. Peixoto showed in [7] that any two Morse-Smale flows on a closed manifold can be joined by an arc with finitely many bifurcations. Discrete systems are in contrast to this result. For example, S. Matsumoto showed in [5] that every orientable closed surface admits two isotopic Morse-Smale diffeomorphisms which cannot be joined by such an arc. For the dimensions larger or equal to 3 the problem is nontrivial even for the simplest diffeomorphisms “north pole - south pole”. The classical result of J. Cerf [3] states that for every two orientation preserving diffeomorphisms (and therefore for every two diffeomorphisms “north pole - south pole”) on \(\mathbb {S}^3\) there is a smooth arc joining them. We show that it can be so chosen that the whole arc consists of the “north pole - south pole” diffeomorphisms. We show that for dimensions larger then 3 the problem is even more complicated. On the sphere \(\mathbb S^6\) there are two “north pole - south pole” diffeomorphisms that cannot be joined by a smooth arc (it follows easily from the result by J. Milnor). Finally we show the way in which the sequence of two saddle-knot bifurcations results in transition from one class of topological conjugacy to another in set of Pixton diffeomorphisms. The main results of this chapter are in the papers [1, 2, 4, 9].

In the classical papers by S. Smale and J. Palis the proof of structural stability of Morse-Smale diffeomorphisms was based on the construction of a system of tubular neighborhoods. We present the similar construction for gradient-like 3-diffeomorphisms using the idea of representation of the dynamics of the system as “attractor-repeller” and the consideration of the space of the wandering orbits. We come to the compatible system of neighborhoods which plays the key role in the topological classification. Let \(MS_0(M^3)\) denote the class of gradient-like diffeomorphisms on the manifold \(M^3\). In this chapter we give the complete topological classification of the diffeomorphisms of this class by means of the topological invariant called the scheme of the diffeomorphism which generalizes the invariants for the Pixton class. The scheme is a simple 3-manifold whose fundamental group admits an epimorphism to the group \(\mathbb Z\) and a system of tori and Klein bottles smoothly embedded into this manifold. The presented results are for the most part from the paper [5]. In the papers [1, 3, 4, 6‐10, 14] one can find topological classification of some special classes of the Morse–Smale diffeomorphisms on 2-manifolds.

In this chapter we state some interrelations between the topology of the ambient manifold \(M^3\) and dynamics of a diffeomorphism \(f\in MS(M^3)\). These relation deal with the number where \(r_{_f}\) is the number of the saddle periodic points and \(l_{_f}\) is the number of the knot periodic points of the diffeomorphism f. In the section 6.1 we construct the topological classification of closed orientable 3-manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves. Such a manifold is either the 3-sphere if \(g_{_f}=0\) or the connected sum of \(g_{_f}\) copies of \(\mathbb S^2\times \mathbb S^1\). We point out one more interrelation between \(g_{_f}\) and the topology of the manifold \(M^3\) if the diffeomorphism f is gradient-like and it has tamely embedded frames of 1-dimensional separatrices. In this case the ambient manifold \(M^3\) admits Heegaard splitting of genus \(g_{_f}\). The results of this chapter are for the most part contained in [1, 2].

Following the ideas of A. Lyapunov C. Conley introduced the notion of a Lyapunov function for a dynamical system (see Definition 7.1). In 1978 he proved the existence of a continuous Lyapunov function for every dynamical system [2]. This result is called the fundamental theorem of dynamical systems. If a Lyapunov function is smooth and the set of its critical points coincides with the chain recurrent set then this function is called the energy function. Very generally smooth flows admit an energy function (see, e.g. Theorem 6.12 in [1]), but it is not true for diffeomorphisms. First results on construction of an energy function (see Definition 7.2) belong to S. Smale. In 1961 [9] he proved the existence of an energy function, which is a Morse function, for every gradient-like flow (i.e. Morse-Smale flow without closed trajectories). K. Meyer [7] in 1968 generalized this result and constructed an energy function, which is a Morse-Bott function, for an arbitrary Morse-Smale flow. The only result of this kind for diffeomorphisms belongs to D. Pixton [8], who in 1977 proved the existence of an energy function, which is a Morse function, for Morse-Smale diffeomorphisms on surfaces. Furthermore, he constructed a diffeomorphism on the 3-sphere (we have already mentioned it in Chapter 4 as the Pixton’s example) which has no energy function, and he explained the phenomenon to be caused by the wild embedding of the separatrices of the saddle points. Recently the conditions of existence of an energy function were found in [3‐6]. In section 7.1 we present important properties of a Lyapunov function, which is a Morse function, for Morse-Smale diffeomorphisms on n-manifolds. In section 7.2 we introduce a dynamically ordered Morse-Lyapunov function for an arbitrary Morse-Smale diffeomorphism of a 3-manifold with the properties closely related to the dynamics of the diffeomorphism. We show that the necessary and sufficient conditions of the existence of an energy function with these properties are determined by the type of the embedding of the 1-dimensional attractors (repellers), each of which is the union of the 0-dimensional and the 1-dimensional unstable (stable) manifolds of the periodic points of the diffeomorphism.

In this chapter we study orientation preserving A-diffeomorphisms on an orientable compact manifold \(M^n\) (possibly with boundary) with a nontrivial basic set \(\varLambda \) in the interior of \(M^n\). We state some important properties of the basic sets in relation to their type and dimension. These properties are used for the topological classification of the basic sets (including expanding attractors and contracting repellers) as well as for important classes of structurally stable diffeomorphisms. We present the constructions of classical A-diffeomorphisms with basic sets of codimension one: the DA-diffeomorphism, the diffeomorphism with the Plykin attractor, the diffeomorphism with the Smale “horseshoe”, the diffeomorphism with the Smale-Williams solenoid. The results of this chapter are based on [1‐4, 7, 10, 13‐20].

The first problem which arises when studying a topological classification of A-homeomorphisms with nontrivial basic sets is the problem to find a suitable topological invariant which can adequately reflect the restriction of the diffeomorphism to the basic set as well as the embedding of the basic sets to the ambient manifold. Let \(f,~f'\) be orientation preserving A-diffeomorphisms of orientable compact manifolds \(M^n,~{M}^{\prime n}\) (possibly with boundary) respectively and let \(f,~f'\) have nontrivial basic sets \(\varLambda ,~\varLambda '\) respectively which are in the interior of the manifold. The problem is to find necessary and sufficient conditions of existence of a homeomorphism \(h:M^n\rightarrow M^{\prime n}\) such that \(hf|_{\varLambda }=f'h|_{\varLambda }\). We show that the problem of topological conjugacy for arbitrary 2-dimensional and 1-dimensional basic sets as well as 0-dimensional basic sets without pairs of conjugated points can be reduced to the analogues problem for widely situated basic sets on an orientable surface with boundary (possibly empty) which is called a canonical support of the basic set. The study of the latter is essentially based on the investigation of the asymptotic properties of the stable and unstable manifolds of the points of the basic sets on the universal cover. The suggested method solves the problem of realization of arbitrary 1-dimensional basic sets as well as 0-dimensional basic sets without pair of conjugated points by constructing a so called hyperbolic diffeomorphism on the support of the basic set such that it has an invariant locally maximal set on the intersection of two geodesic laminations. We show that the restriction of the diffeomorphism f to the basic set is a factor of the restriction of the hyperbolic diffeomorphism to the intersection of the respective laminations. If the basic set is widely situated on the torus then for it a hyperbolic automorphism of the torus (Anosov diffeomorphism) is uniquely defined and it is a factor of the initial diffeomorphism. If the diffeomorphism f is structurally stable then the support of the basic set can be constructed in such a way that the restriction of the initial diffeomorphism to it consists of exactly one nontrivial basic set and of finitely many hyperbolic periodic points belonging to the boundary of the support. This and the results on the classification of the Morse-Smale diffeomorphisms enable us to construct a complete topological invariant for important classes of structurally stable diffeomorphisms on surfaces whose non-wandering sets contain a nontrivial 1-dimensional basic set (attractor or repeller). Such a class was studied for instance in [8]. The presentation of the results in this chapter follows the papers [2, 6‐12, 16, 17] and it is in many concepts related to the papers [1, 4, 5, 14, 15].

The theory of dynamical systems extensively uses a lot of concepts and tools from other branches of mathematics: topology, algebra, geometry etc. In this chapter we review the basic definitions and facts, necessary for understanding the presented results. We begin the fundamental notions of a set and of a map, we describe which structures one has to define on a set to make it a group, a linear space, a metric space. We recall the main properties of maps and their spaces. We give some facts on embedding of a surface into a 3-manifold and the definition of a wild embedding. We show how the universal cover is constructed and the connection between Nielsen-Thurston’s theory and structurally stable diffeomorphisms of surfaces.