Relative Nielsen Numbers, Braids and Periodic Segments

1 Introduction

The notion of periodic segments introduced by Srzednicki proved to be a very useful tool for detecting periodic solutions and chaotic dynamics generated by periodic in time non-autonomous ODEs [23, 24]. If v:ℝ×ℝn→ℝn is a smooth time-dependent vector field on ℝn, then the ordinary differential equation

generates a local flow on the extended phase space ℝ×ℝn. If f is T-periodic with respect to time, then the Poincaré map P associated to the vector field v (defined on some open subset of ℝn) is given as follows: P⁢(x0) is the value of the solution of the problem

at time T. The k-periodic points of the Poincaré map P correspond to the kT-periodic solutions of x˙=v⁢(t,x).

The periodic segment W is a compact subset of [0,T]×ℝn with some special behavior of the vector field (1,v) on the boundary of W. Namely, the set of points (called the exit set W-) on the boundary of W at which the vector field (1,v) is pointing out with respect to the segment is closed. Periodicity of the segment means that the time 0 and time T sections of W are equal, W0=WT. Moreover, there exists a compact set W--⊂W- (called the essential exit set) such that W0--=WT--, and (W,W--) is a pair of trivial bundles over [0,T] with the fiber (W0,W0--). Intuitively, W consists of the left-hand side {0}×W0, the right-hand side {T}×WT={T}×W0, and the main part located over the open interval (0,T). Because of the specific behavior of the flow (it moves along the time-axis with speed 1), it is clear that the right-hand side of W must belong to the exit set (see Figure 1).

There exists a relative homeomorphism mW:(W0,W0--)→(W0,W0--) associated to the periodic segment W, called the monodromy map. It was proved by Srzednicki [24] that there exists an isolated set of fixed points Fk of Pk contained in W0∖W0-- such that

where m:=W0∋x→mW⁢(x)∈W0, m¯=m|W0--. In particular, if L⁢(mW)≠0, then the Poincaré map has a fixed point x whose trajectory (with respect to the local flow on the extended phase space) is contained in the segment W. The proof of Srzednicki’s fixed point index formula is based on the Lefschetz fixed point theorem and the properties of the fixed point index.

Figure 1

The periodic segment W over [0,T] with the essential exit set W--.

The problem of using other topological invariants to get more information on the structure of periodic solutions inside the segment was proposed in [22]. In addition to the existence of fixed points of the Poincaré map P, we are interested in finding a lower bound of the cardinality of the set of fixed points Fix⁡(P). From that point of view, it seems to be very natural to try to find a connection between the method of periodic segments and the theory of Nielsen numbers (see [26]). The Nielsen fixed point theory is concerned with the determination of the minimal number of fixed points for all maps in the homotopy class of a given self map f:X→X (X is a compact ENR). The Nielsen number N⁢(f) provides a homotopy invariant lower bound for the number of fixed points of f.

The classical Nielsen number N⁢(f) is rather poor lower bound for the number of fixed point of f if f:(X,A)→(X,A) is a relative map. For example, let X be a 2-dimensional disk and let A be the circle bounding it. If f:(X,A)→(X,A) is a continuous map such that f¯:=f|A has degree d, then f has at least |d-1| fixed points in A. On the other hand, N⁢(f)=1. In 1980, Jiang observed the following phenomena in the problem concerning fixed point sets on the pants. Let P be the pants, i.e., the disk with two holes removed, and let f be the homeomorphism obtained by reflection on an axis of symmetry which interchanges the boundaries of the two holes. Then N⁢(f)=1, but any homeomorphism isotopic to f will map the outer boundary of P onto itself in an orientation-reversing manner, and hence have at least two fixed point on this boundary circle. Thus, N⁢(f) cannot be realized by a homeomorphism in the isotopy class.

In 1986, an extension of Nielsen theory to the relative setting was introduced by Schirmer in [19] and has developed rapidly since then ([31] is a very interesting survey on the subject). In this paper we will use the relative Nielsen number of f:(X,A)→(X,A) on the closure of the complement N⁢(f;X∖A¯), defined by Zhao [31, 30, 29].

We show (Theorem 4.4) that if W is a periodic segment over [0,T], then the Poincaré map P has at least N⁢(m,W0∖W0--¯) fixed points with trajectories contained in the segment W. We give an example showing that N⁢(m,W0∖W0--¯) cannot be replaced by the relative Nielsen number N⁢(m;W0,W0--), defined by Schirmer. We will also study the relation between the number of periodic points of the Poincaré map P and the asymptotic Nielsen number N∞⁢(m) defined and developed by Jiang (see [14, 11, 12]). We prove (Theorem 5.2) that if the sequence N⁢(m¯n) is bounded and N∞⁢(m)>1, then the Poincaré map has infinitely many periodic points. We also show (Theorem 5.3) that there exists a compact set I⊂W0, invariant for the Poincaré map, such that the topological entropy h⁢(P|I) is bounded from below by log⁡N∞⁢(m)-h⁢(m¯). In particular, if h⁢(m¯)=0, then

The usefulness of our main results (Theorems 5.2 and 5.3) depends on the possibility of computation of N∞⁢(m). The setting in which a great work has been carried out is that of homeomorphisms of compact surfaces. The central area in the topological dynamics of surface homeomorphism is their classification up to isotopy, due to Nielsen and Thurston (see [13, 17, 25]). Given a homeomorphism f (relative to some given finite f-invariant subset A) of a compact surface X, perhaps with boundary ∂⁡X, there exists a canonical Thurston representative g in the isotopy class of f that is one of the following three types: finite order (so gn=id for some n≥1), pseudo-Anosov, or reducible. In the third case, the surface may be cut up into subsurfaces along a tubular neighborhood of a finite g-invariant set of mutually-disjoint curves, and the restriction of an appropriate iterate of g to each subsurfaces is either finite order or pseudo-Anosov. Given the action of f on the fundamental group π1⁢(X∖A), one can effectively decide its Thurston type using an algorithm due to Bestvina and Handel [2]. If W0 is a compact surface with boundary, then the Thurston–Nielsen canonical form decomposes m into periodic and pseudo-Anosov pieces where there is a stretching factor λ which describes each pseudo-Anosov piece. It was proved by Jiang (see [14, 11, 12]) that if the Euler–Poincaré characteristic W0 is negative, then N∞⁢(m) equals the largest such stretching factor λ>1 (λ:=1 if there is no pseudo-Anosov piece). In particular, if there is at least one pseudo-Anosov piece, then N∞⁢(m)>1.

As a natural possible area of applications, we will study braid-like periodic segments for periodic in time ODEs on the plane ℝ2. Using the technique developed in [13, 14], we give an example of a periodic braid-like segment with N∞⁢(m)>1. It is based on result of Fadell and Husseini (see [6]), using the Fox’s free differential calculus.

The study of geometric braids in the context of the existence of periodic solutions of periodic system of differential equations on the plane was started by Matsuoka in [15, 16, 18, 17] (see also [1, 13]). Based on the braiding information of known solutions, he obtained lower bounds for the number of extra periodic solutions.

This approach has one failing from the point of view of the applications to dynamics generated by ordinary differential equations. It is assumed that every solution of the equation x˙=v⁢(t,x) extends forever in both directions of time. In particular, the Poincaré map P has to be defined globally as a diffeomorphism P:ℝ2→ℝ2. Matsuoka’s method applies also to the case of dissipative systems, when the Poincaré map P has a closed disk D such that P⁢(D)⊂D. It is rather a rare phenomenon in the concrete examples of ODEs. The advantage of our approach lies in the fact that we consider the local flows, so the solutions can blow up to infinity in finite time and the Poincaré map does not have to be globally defined. Moreover, we allowed the periodic segments with the non-empty essential exit sets. It should be stressed that Matsuoka’s approach and the method described in [14, 11, 12] cannot be directly applied to study the number of periodic points of the Poincaré map restricted to W0 if W0--≠∅. Our viewpoint sheds some new light on the structure of periodic orbits in that more general case.

We emphasize that if W0 is contractible, then the Nielsen numbers N⁢(mn,W0∖W0--¯) do not give more information on the number of periodic points than the Lefschetz numbers L⁢(mWn). We show that in some examples of chaotic planar periodic ODEs, the number of k-periodic points of the Poincaré map is bounded from below by |L⁢((I-μW)k)|, where μW is induced by mW in homologies.

The paper is organized as follows. In Section 2 we have compiled some basic facts concerning Nielsen fixed point theory. In Section 3 we introduce the notion of a periodic segment. Section 4 establishes the relation between the method of periodic segments and Nielsen theory. In Section 5 will be concerned with the topological entropy of the Poincaré map. Section 6 is devoted to studying braid-like periodic segments. In Section 7 we give an example of a braid-like periodic segment that forces complicated dynamics. Section 8 deals with the case of contractible W0. For the convenience of the reader, in Appendix A we give a brief survey of the used results in the Nielsen theory based on [13].

2 Relative Nielsen Numbers

In this section we summarize without proofs the relevant material on the Nielsen theory. Let X be a compact ENR and let f:X→X be a continuous map. The fixed point set of f,

splits into a disjoint union of fixed point classes – two fixed points are in the same class F if and only if they can be joined by a path which is homotopic (rel end-points) to its own f-image. More precisely, x0,x1∈Fix⁡(f) are in the Nielsen relation if there exists a continuous map α:[0,1]→X such that α⁢(i)=xi for i=0,1 and

Then

where Fi are Nielsen fixed point classes. Each Nielsen class is an isolated set of fixed points of f, so the fixed point index ind⁡(f,Fi)∈ℤ is defined. A fixed point class F is called essential if ind⁡(f,F)≠0. The Nielsen numberN⁢(f) of f is defined as the number of essential fixed point classes. Every map homotopic to f has at least N⁢(f) fixed points.

Let f:(X,A)→(X,A) be a continuous map of the pair (X,A) of compact ENRs. The restriction of f to A is written as f¯:=f|A:A→A.

By N⁢(f,f¯) we denote the number of common and essential fixed point classes of f. It follows that

Let f:(X,A)→(X,A) be a relative map. Assume that A1,…,Ak are all components of A such that f⁢(Ai)⊂Ai (i=1,…,k). Then another relative number N′⁢(f,X∖A¯) is defined (see [31]) as the sum of the number of essential fixed point classes of all fAi:Ai→Ai with int⁡(Ai)=∅ and the number of the fixed point classes which do not assume their indices in A and do not contain any essential fixed point classes of fAi:Ai→Ai with int⁡(Ai)=∅. One can prove (see [31]) that N′⁢(f,X∖A¯)≥N⁢(f,X∖A¯).

We define the following auxiliary numbers:

1. N ⁢ ( f , A ) is the number of essential fixed point classes of f which assume their index in A. Obviously, 0≤N⁢(f,A)≤N⁢(f,f¯).

2. n ⁢ ( f ; X , A ) is the number of fixed point classes of f which do not assume their index in A and are a common point class of f and f¯. Obviously, n⁢(f;X,A)≤N⁢(f,X∖A¯).

It follows that

By [21, Theorem 3.4], we have

3 Blocks and Periodic Segments

In this section we recall the notion of a Ważewski set and a periodic segment. We begin with the definitions of the basic concepts of the theory of continuous-time dynamical systems.

Let X be a topological space. A local semiflow on X is a continuous map ϕ:D→X, where D is an open subset of X×[0,∞), such that for every x∈X, the set {t∈[0,∞):(x,t)∈D} is equal to an interval [0,ωx) for some 0<ωx≤∞. If t∈[0,ωx), then ωϕt⁢(x)=ωx-t and the following equations hold:

Let ϕ be a local semiflow on X. For B⊂X we define its exit set by

Let another subset of B be defined as

We call B a Ważewski set for ϕ if B and B- are closed. A compact Ważewski set B is called a block.

We will use the following notation: π2:ℝ×X→X and π1:ℝ×X→ℝ are projections. For Z⊂ℝ×X and t∈ℝ, we define the t-section of Z by

By a local semi-process on a topological space X we mean a continuous map Φ:D→X, where D is open subset of ℝ×X×[0,∞) such that the map

is a local semiflow on ℝ×X. In the sequel, we write Φ(σ,t)⁢(x) instead of Φ⁢(σ,x,t). We say that Φ is T-periodic if Φ(σ,t)=Φ(σ+T,t) for each σ and t.

Figure 2

The periodic segment W and the homeomorphism h preserving t-sections.

Figure 3

The monodromy map m:W0→W0 for some periodic segment Wover [0,T] induced by some homeomorphism h:[0,T]×W0→W.

Let Φ be a local T-periodic semi-process on X and let ϕ be the corresponding local semiflow on ℝ×X. A set W⊂[0,T]×X is called a periodic segment over [0,T] if it is a block with respect to ϕ such that the following conditions hold:

1. There exists a compact subset W-- of W- (called the essential exit set) such that

   W - = W - - ∪ ( { T } × W T ) , W - ∩ ( [ 0 , T ) × X ) ⊂ W - - .

2. W 0 = W T , W0--=WT--, where W0 and W0-- are compact ENRs.

3. There exists a homeomorphism h:[0,T]×W0→W such that π1∘h=π1 and

   h ⁢ ( [ 0 , T ] × W 0 - - ) = W - - .

For the periodic segment W over [0,T] one can define the corresponding monodromy map (see Figures 2 and 3)

The monodromy map is actually a homeomorphism. We will use the following notation:

For s∈[0,T], we define two auxiliary functions by

It follows that

Figure 4

The periodic segment over [0,2⁢π] for the 2⁢π-periodic equation z˙=z¯⁢ei⁢t, z∈ℂ.The essential exit set W-- is shaded. The monodromy map is a rotation by π.

4 Periodic Segments and Relative Nielsen Numbers

Let W be a periodic segment over [0,T] for the T-periodic local semi-process Φ on X and let P=Φ(0,T) be a Poincaré map. Assume that σ:W*→[0,+∞) is the exit time map. Since W*=W, σ is defined and continuous on the whole segment W.

We define a map w:W0→W0 by

Observe that

so we can treat w as a relative map, w:(W0,W0--)→(W0,W0--).

Figure 5

The set W0 is a square with three holes. The open set V={x∈W0:σ⁢(0,x)<T}⊂W0--is shaded in grey. On the complement W0∖V the map w coincides with the Poincaré map P.

Figure 6

The T-periodic segment for z˙=z¯, z∈ℂ. The essential exit set W-- is shaded. The monodromymap mW is the identity, so N⁢(mW;W0,W0--)=2, and the Poincaré map P has exactly one fixed point.

5 Periodic Points by Periodic Segments

Let Φ be a T-periodic (T>0) semi-process on X. We define an operation of gluing periodic segments. If W and Z are periodic segments over [0,T] having the same cross-section at 0, i.e.,