Remarks on Braid Theory and the characterisation

Remarks on Braid Theory and the characterisation of periodic orbits Mario A. Natiello Dept. of Quantum Chemistry Uppsala University Box 518, S-751 20 UPPSALA, Sweden

and Hernán G. Solari Dept. de F´ısica, Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires Pabell´ on I, Ciudad Universitaria, 1428 Buenos Aires, Argentina

April 5, 1994 Abstract The relationship between Braid Theory and the organisation of periodic orbits of dynamical systems is considered. It is shown that for some (physically relevant) 3-d flows the characterisation of periodic orbits by means of Braid Theory can be done on the Poincaré surface in an efficient way. The result is a thread-less graphical presentation of a braid class. We discuss extensions of this approach to (adequate) dynamical systems of dimension higher than three, using results from Central Manifold Theory.

I

Introduction

The comparison of models with data, with the consequent rejection or validation of the model, is a fundamental problem of physics and natural sciences in general. In this context, the straightforward comparison of measured time-traces with their modelled counterparts has been the classical validation test. However, in the presence of chaotic dynamics, this technique is inadequate due to the divergence of nearby trajectories which is characteristic of the dynamics. Even the notion of reproducibility of experimental results —the “proof” of experimental science— has to be reconsidered in the context of chaos, since the sensitivity to initial conditions renders unthinkable the long-time agreement of the time-trace of two different runs of the experiment under “identical” external conditions. The methods used to validate models working in the chaotic regime have to rely on techniques that capture the organisation of the strange attractor as a whole. Such a characterisation, or classification, is still needed in the general case. In the case of problems that admit a model based on a 3-dimensional dynamical system, the structure of the periodic orbits contained in the invariant set is of importance to understanding the organisation of the flow. Properties such as relative rotation rates, linking numbers and the like [1], 1

that are persistent under small perturbations of the dynamical system have been used in the past as the basis for comparison. So far, all attempts at analysing the organisation of periodic orbits have been done on the flow, i.e. following periodic orbits throughout time either by numerically solving the differential equations of the model[1] or by manipulation of measured data from the system in order to obtain a representation of the phase-portrait[2, 3, 4]. There are many examples of 3-dimensional systems having a strange attractor that in addition admits a (local) Poincaré section [5, 6, 7, 8] and its associated first-return or Poincaré mapping. It is known that the Poincaré mapping is a discrete counterpart of the flow preserving all relevant dynamical information (note that global torsions by multiples of 2π in the flow cannot be detected on the Poincaré section). However, previous work in orbit organisation was based on properties of the flow. A second goal in identifying the organisation of periodic orbits is to make contact with the work based on Thurston’s classification theorem for diffeomorphisms of surfaces [9], which allows the extraction of dynamically relevant information from knowledge of the braid type of the periodic orbits. From this knowledge it can be said whether the periodic orbit structure implies the existence of Smale’s horseshoes [10], or conversely the compatibility of the model or data with a diffeormophism with zero entropy [11] and even the estimation of a lower bound for the topological entropy [12, 13, 14]. The aim of this work is two-fold. In the first place, to show that the characterisation of periodic orbits by means of Braid Theory can be done on the Poincaré surface in an efficient way. This novel — to our knowledge — presentation of the braid group is curiously thread-less: In the standard approach, the “stuff” that is plaited when associating an orbit to an element of the braid group is the flow itself. The association is here performed on the Poincaré surface instead. Strictly speaking, the presentation is faithful up to a global torsion, i.e. on the quotient group between the braid group and the (normal) subgroup describing global torsions. During the process of reviewing we have come in contact with a closely related result by T. Hall [15], see Section II.2. Secondly, we discuss a topic concerning the existence and relevance of braids and knots, namely the fact that knots and braids disappear in dimensions higher than three when considered as embeddings of a circle, vs. their persistence through their association with the dynamics. In fact, 3-d models for physical systems frequently arise as an approximation for more accurate higher-dimensional models (for instance reduction of partial differential equations[16]). It is then reasonable to question if a periodic orbit on, e.g. a four-dimensional phase-space can build a knot or braid at all. The organisation of the paper is as follows: In Section II we discuss the presentation of the braid group on the Poincaré surface from an intuitive point of view. The proof is discussed in Section III. Section IV addresses the problem of higher dimensions. We defer to the appendices most of the proofs or sketches of proofs of the statements presented in the article.

II

Braids, knots and periodic orbits

In this section we will discuss intuitively the connection between the braid group and a group structure induced by closed curves in R2 on π0 (Yn (x0 ), f0 ) (i.e. the path-components (π0 ) of the set {Yn (x0 ), f0 }, to be defined below). This connection provides a graphical description of the braid group modulo the global torsion. The proof that both group structures are isomorphic up to global torsions will be given in the next section. We are used to visualising orbits in 3-d flows as knots, links and also as braids[2, 11, 17] if a Poincaré section can be identified. In this case, one can construct the first return map, F : R2 → R2 , which contains almost all the dynamical information. A period-n orbit in this map (i.e. n different points that are cyclically mapped among themselves) defines a period-n closed orbit in the flow. Earlier work 2

on Poincaré maps[18] supports the claim that (orientation preserving) 2-d maps and 3-d flows with a (global) Poincaré section are equivalent in their dynamical properties, up to a global torsion. However, when facing the practical reconstruction of orbit organisation we have to resort to suspensions. Do we really need them? A practical remark is that it is harder (and sometimes impossible) to get a good reconstruction of the flow while, at times, it is not so hard to get a good reconstruction of the return map[2, 3]. For the return map we can easily avoid the regions where the flow develops in the plane perpendicular to the measured variable —which is precisely the region where the crossings are difficult (if not impossible) to reconstruct. Ideally, both reconstructions (flow and map) will be complementary; for example, in identifying close returns the flow reconstruction (in a large time-delay embedding) is more robust than the orbit search in the Poincaré section. In terms of Poincaré maps, a possibility for identifying the braid associated with a set of periodic orbits is to rely on the definition [19] of braids in terms of the fundamental group of a punctured disk, finding the image of the loops generating the group through the Poincaré map. This method will be very demanding for relatively large sets of periodic orbits.

II.1

Braids and Knots

Artin [19] introduced braids pictorially, see Fig. (1). Each braid is made of n-strands going from the top surface to the bottom surface (in our case the surface represents the Poincaré surface). Two braids can be composed by joining them “bottom-to-top” and erasing the middle line, Fig. (2). The resulting object is a new braid. Thus, braids are equipped with a product and form a group under this operation, the identity being the braid with n-strands without crossings. The pictorial presentation can be made algebraic in terms of rules imposed on the generators σi of a free group. Following Artin we define the elementary braid σi as the crossing of the i-strand with the (i+1)-strand when the i-strand goes over the (i+1)-strand, Fig. (3). The inverse of this operation is σi−1 and is also depicted in Fig. (3). It is clear that we can create any braid by composing the elemental moves and that we can label each pattern according to the string of σi ’s required to create it from the identity. In so doing, we can verify that different labellings (names) can give rise to the same pattern. In order to make the relation between names and patterns bijective we identify the braids σi σi+1 σi = σi+1 σi σi+1

(1)

σi σj = σj σi if |j − i| > 1

(2)

as partially illustrated in Fig. (4). The (equivalent) definition of the braid group we will consider is[20, pp. 21-31]: Bn = π1 (R2n − ∆)/P (n)

(3)

Bn is the group of braids of n strands. π1 (X) is the fundamental group of X, X a topological space. R2n stands for n copies of R2 and ∆ is the great diagonal of R2n , i.e. the n- tuples (x1 , · · · , xn ) such that there exist i 6= j with xi = xj . If the whole (R2n − ∆) is considered (i.e. not taking the quotient space with the permutation group P (n)), one obtains instead the group of coloured braids of n strands, Pn . This group contains those braids that after acting on the n strands leave their ordering unchanged. Intuitively, the difference between coloured and colourless braids amounts to joining the n points describing the strands in any possible ordering. There is an exact sequence[20, 21] connecting Pn and Bn , namely: 3

1 → Pn → Bn → P (n) → 1

(4)

Braids relate to knots, links and periodic orbits on a 3-d flow: identifying points in the upper and lower ends of the braid patterns (again, see Fig. (3)), a braid pattern can be associated to a periodic orbit (or to a set of periodic orbits). This procedure is clearly many-to-one since any braid b will produce the same knot and periodic orbit as any other braid w, conjugated to b, i.e. w = vbv −1 for some v. One actually associates orbits with braids up to conjugation, or equivalently, classes of braids with periodic orbits. In terms of flows, conjugation can be seen as a change in the projection and/or a change between equivalent Poincaré surfaces. Moreover, the integer powers v k , where v = sn and s = (σ1 · · · σn−1 ), themselves form a group (isomorphic to the additive group of the integers, Z) which is a normal subgroup of the braid group; this group is the center of the braid group, Z(Bn ) [19]. A global torsion is described by a fixed power k of v. This represents a global 2kπ-rotation of the whole flow around itself. Clearly, the association between classes of braids and periodic orbits on the Poincaré surface cannot recognise global torsions. Knots and periodic orbits are not exactly the same objects. One has more freedom to disentangle a knot than an orbit, the reason being that in the second case the periodicity of the orbit (with respect to the Poincaré section) has to be preserved, while in the knot case the concept of periodicity is meaningless. This can be illustrated by comparing a period two orbit with a period one orbit, Fig. (5). Both orbits are unknots. In terms of the algebraic presentation we can go from the period one orbit to the period two orbit by adding one strand and multiplying by σ2 . In terms of knots these two braids are equivalent.

II.2

Links in the plane

The question we address in this subsection is: given a Poincaré map F and a periodic orbit of (minimum) periodicity p, X = {xi , i = 1, ..., p}, how can we determine the associated braid? More exactly, we are interested in the quotient group between the braid group and the global torsion subgroup. The construction turns out to be very simple. Consider each point xi as the location of a nail. Connect each nail with the next one following the order of any permutation of the sequence {ij }, j = 1, · · · , p and avoiding crossings. Connect finally the last nail with the first one. Note that this construction can be realised by a continuous injective loop on the Poincaré surface joining the points of the periodic orbit. The braid acts as an operator over this structure. First, find the proper representative of the braid by conjugating b in such a form that in the top surface the points are ordered in the same sequence as the ij . Second, find the σ -representative of this braid. Translate the action of b as an operator on the structure. In order to perform the last operation observe that the induced deformation of σj consists in permuting the nails ij and ij+1 rotating counter-clockwise without detaching any string or lifting the nails off the surface. The procedure is illustrated in Fig. (6). Thus, in order to recognise the braid associated with a periodic orbit X of a map F we have to find the image of a simple closed curve passing through the points of X. Clearly, a global rotation of the whole flow will pass unnoticed, so this procedure can only represent braids up to a global torsion. This presentation results in the statement: given a 2-d Poincaré map, F , of a disk into a disk and a set of periodic orbits of F , it is possible to establish the braid structure of the set of periodic orbits up to conjugation. In practical terms it means that the braid structure and with it linking numbers, relative rotation rates, knot polynomials, etc. can be reconstructed if one is at least able to find a good representative for a Poincaré section and its associated first return map as well as representatives of the periodic orbits, even if one is not able to faithfully reconstruct the 3-d flow. 4

Note that since we are associating braids with the images by F of a certain type of closed loops on a 2-d region (the exact definition of this set of loops, M, will be discussed in the next section), we have to demand F to preserve the properties of the loops. It only remains to be shown that the group of transformations of closed curves on the Poincaré section actually is isomorphic to the braid group quotient by the subgroup of global torsions. This will be discussed in the following Section. Finally, we note that presentations related to the present one have been used in the past to illustrate braids [10, 22]. In particular, T. Hall [15, p. 65] discusses a similar construction (named the p-line diagram) in the context of Thurston’s theory, proving that two braids producing the same p-line diagram belong to the same class.

III

The proof

The aim of this Section is to describe in a more formal language the correspondence between classes of braids (describing sets of periodic orbits) and loops in the plane. To this end, we will highlight the different steps of the proof in this section, delaying the details to the appendices. Consider a set of n different points in R2 , (x1 , · · · , xn ), which can be assumed to lie on a closed, smooth curve without self- intersections. These points can be interpreted as a collection of n points belonging to a set of periodic orbits on a 2-d Poincaré section as discussed in the previous section (n will be related to the braids of n strands). An ordering for the points can be defined by moving along the curve, say counterclockwise, and choosing a starting point arbitrarily. Consider now the pointed sets (M, f0 ) = ({f : S 1 → R2 , f continuous, injective, counterclockwise and bounded }, f0 ) and

(Xn , x0 ) = ({R2n − ∆}, x0 )

where f0 is a distinguished embedding of the circle in R2 . Xn represents all n-tuples of points in R2 , where xi 6= xj for i 6= j, with the selected point x0 = (x1 , · · · , xn ). ∆ is the grand diagonal of R2n i.e. the n-tuples where two or more components coincide. The space M can be equipped with the supremum norm when necessary. We define a projection pn : M → Xn via pn (f ) = (f (θ1 ), · · · , f (θn )) , 1

where θi ∈ S , θi < θi+1 . The choice of the set {θi } has to be made once for all times at the beginning of the procedure, so that each f has a unique projection p(f ). The point (x1 , · · · , xn ) with f (θi ) = xi , i = 1, · · · n belongs to Xn since f is simple and θi 6= θj for i 6= j. Finally, the triple (M, Xn , pn ) defines a fibration with fibre Yn (x) = {f ∈ M : f (θi ) = xi ,

i = 1, · · · , n}.

(5)

We note that different choices of θi , will lead to equivalent (for our purposes) fibrations. The fibre at a point x consists of all functions in M passing through the points xi in the proper order. π0 (Yn (x)) will denote the equivalence classes of these functions with respect to homotopies. The homotopy equivalence classes can be described intuitively as the classes of continuous, non intersecting, closed curves in R2 passing through the selected points xi of R2 that can be continuously deformed 5

into each other without crossing these selected points. In appendix A we discuss the fact that this triple actually is a fibration. The main result we want to prove, is the following: Theorem 1 π0 (Yn (x)) is isomorphic to the quotient between the braid group π1 (Xn , x0 ) and the subgroup of global torsions. This result follows from the property known as the exact sequence of the fibration[21]. The relevant portion of this sequence is: · · · → π1 (M, f0 ) → π1 (Xn , x0 ) → π0 (Yn (x0 )) → π0 (M, f0 )

(6)

where π1 (Xn , x0 ) is the (coloured) braid group[20]. The whole argument holds equally well if instead of (Xn , x0 ) we consider its quotient space with the permutation group P (n). We obtain thus the group Bn of colourless braids of n strands, instead of the group of coloured braids, as discussed in the previous section. The group Bn is in fact the most important concerning the description of dynamical systems. Unless otherwise stated we will in the following adopt the colourless braids as the “default” situation. To avoid extra complications in the notation, we will assume that (Xn , x0 ) denotes already the quotient space with P (n). The isomorphism is a consequence of the following facts: 1. The sequence (6) is exact. This is immediate after proving the fibration property (see Lemma 1). 2. π0 (M, f0 ) = {0}, i.e. M is path-connected, see Lemma 2. 3. π1 (M, f0 ) ' Z. See Lemma 3. φ

4. The mapping π1 (Xn , x0 )/p∗ (π1 (M)) → π0 (Yn (x0 )) induces a group structure on π0 (Yn (x0 )) by exactness. Here, p∗ stands for the map on homotopy classes induced by the projection p on π1 (M), see Lemma 4. The procedure is the following. First we characterise the relevant part of the exact sequence in Lemmas 2 and 3. Next we observe that the image of π1 (M) in π1 (Xn , x0 ) induced by the projection p is a normal subgroup of the braid group (i.e. the global torsions) and we “quotient out” this normal subgroup in the exact sequence. This is done in Lemma 4. By exactness it follows that the mapping φ that induces the group structure on π0 (Yn (x0 )) is both surjective and injective. This induced group structure is then isomorphic to that of the quotient group π1 (Xn , x0 )/p∗ (π1 (M)). This completes the proof. Lemmas 1 to 4 are considered in Appendix A. The fact that the Poincaré map induces a map in π0 (M, f0 ) allows one to associate an element of the braid group (quotient with global torsions), π0 (Yn ), with a set of periodic orbits of the Poincaré map by choosing an element in M with projected points xi = f (θi ) such that the points {xi } belong to the periodic orbits of the first return map, making connection with Section II.

IV

Braids and higher-dimensional dynamical systems

Our primary goal has been to find a practical way of identifying automorphisms (maps), F , in R2 through the braids associated to their periodic orbits. Clearly, every continuous injective map F is also an automorphism in M. If, in addition, the point x0 in the base space Xn is built from (complete) periodic orbits of F , the map F induces a map in the fiber of x0 and we are entitled to associate with the pair (F, x0 ) an element of the group of braids, achieving the desired goal. 6

When dealing with applications, the following concern usually appears: “In general, 2-d maps (3-d flows) arise as extreme models or reductions from higher dimensional models of the problem. It is also known that knots can be unknotted in dimension four and higher. Then, are the braids (knots) representatives of the dynamics or, in opposition, artifacts of the description?”. We ought to expect that if the reduction from high to low dimensionality carries any meaning, then the braids in the low-dimensional flow will have a counterpart in the higher dimensional system. A partially affirmative answer to this conjecture can be proved as a corollary of a modified version of the Center Manifold Theorem, which might be called the Adiabatic Elimination Theorem. We formalise the discussion in the next subsections.

IV.1

The center manifold theorem (flows)

Let us first recall the Center Manifold Theorem (CMT) in the version adapted to global invariant manifolds for singular perturbation problems (flows). Theorem 2 Global Center Manifold (flows)[23, p. 31] Consider the system x0 y0 0

= Ax + f (x, y, ) = By + g(x, y, ) = 0

(7)

(the 0 indicates time derivatives) where x ∈ Rl , y ∈ Rm and A and B are matrices whose eigenvalues have zero and negative real parts respectively. Suppose also that f , g are C 1 with f (0, 0, 0) = 0, g(0, 0, 0) = 0. Let M > 0. Then, there is a δ > 0 such that (7) has an invariant manifold y = h(x, ), |x| < M , || < δ, with |h(x, )| < C||, where C is a constant which depends on M, A, B, f and g. The flow in the center manifold follows the dynamics given by x0 = Ax + f (x, h(x, ))

(8)

Finally, orbits starting sufficiently close to the center manifold will converge to it exponentially fast, i.e. for z sufficiently small, the orbit with initial condition (x0 , h(x0 ) + z) differs in O(e−γt ) from a solution of (8), where γ > 0 depends only on B. Let us consider only the case in which A is identically zero. It is then convenient to scale the time with , i.e. t = τ producing a flow on the center manifold described by the equation (see for example [23, p. 8], [24]): x˙ = f (x, h(x, )) (9) Where the dot denotes the τ -derivatives. Note that the vector field f in (9) stands for any sufficiently smooth and bounded (in compact domains) vector field and has no fundamental restriction in the dynamics it displays. The choice of the τ -scale makes this fact more evident.

IV.2

Center Manifolds for Maps

The map counterpart of the global Center Manifold Theorem can be formulated and proved starting from the local version as can be found in e.g. [25] and paralleling the discussion of [23, pp. 16, 30] for the local and global flow-theorems (see Remarks below). We present here a slightly different version instead.

7

Theorem 3 Consider the map xn+1 yn+1

= f (xn , yn , ) = g(xn , yn , )

(10)

x ∈ Rk , y ∈ Rm , f (x, y, ) and g(x, y, ) are C 1 -Lipschitz continuous functions, g(0, 0, 0) = 0 = f (0, 0, 0). φ0 is a C 1 -Lipschitz diffeomorphism in Rk given by φ0 (x) = f (x, 0, 0). Given M > 0, for sufficiently small and |x| ≤ M , there is a “Global Center Manifold” for (10) of the form (x, y) = (u, h(u)) where the evolution is given by the map un+1 = f (un , h(un ), )

(11)

The center manifold is globally attractive and the orbits (x, y) = (u, h(u) + z) decay faster that some γ n (γ < 1) towards the orbit of (u, h(u)) for sufficiently small z and . We defer all the technical aspects of the proof to Appendix B. In the limit case of = 0, the center manifold coincides with the x-plane while the map is singular and loses memory of the y component in just one step. The theorem simply establishes that this situation persists for small enough , i.e. that an invariant manifold exists (the center manifold) that is a smooth deformation of the x-plane and that trajectories outside this invariant manifold are rapidly “squeezed” towards the center manifold. Remarks: 1. Note that the correspondence between the map in the center manifold and the full map is established for all (small enough) . This is not to say that maps for different values of will be conjugated. 2. It is customary to require f, g to be C 2 instead of C 1 - Lipschitz, although this does not alter the general strategy of the proof [25, p. 44].

IV.3

Braids for n-dimensional maps with a 2-dimensional global stable center manifold

We are now ready to state the main result of this section. In the spirit of Sections I-III, we will be interested in the case l = 3, i.e. when the center manifold is 3-dimensional (and moreover admits a Poincaré section). Hence, the periodic orbits (if present) will form knots and links embedded in the center manifold. Theorem 4 Consider the map (10) for k = 2. Let p be a finite union of periodic orbits of the reduced map (i.e. the map in the center manifold) and P the periodic orbits of the full map dressed with their strongly stable manifold (i.e. the orbits decaying towards the center manifold faster that γ n ). Then, π1 (Xq ) = π1 (Zq ), where π1 (Xq ) is the braid group of q strands, Zq = (R(2+m)q − ∆m q , P ), and mq ∆m = ∆ × R q q In other words Zq is obtained multiplying each element of the q-tuples in Xq by Rm . The index q labels the number of strands in p. The proof follows as a consequence of the stable character of the center manifold. The strongly stable manifold of a periodic set p is the union of q pieces W ss (xk , yk ) = (xk , h(xk ) + z) where k = 1, · · · , q and z ∈ B m , with B m ∈ Rm a sufficiently small cube centered at zero. There is a deformation retract from the strongly stable manifold into the Center Manifold given by (xk , h(xk ) + (1 − t)z) with t varying from 0 to 1. Correspondingly, there is a deformation retract from Xq to Zq . Hence, both spaces have the same homotopy type. This suffices to prove the theorem. 8

The relevance of the theorem resides in that ∪qk=1 W ss (xk , yk ) is invariant under the action of the map, hence we can associate braids with a map in a dimension higher than 2 in a meaningful way. This result is enough to recover the connection between periodic orbits and elements of the braid group as discussed in Sections I-III. The situation presented in this section, although it may look farfetched to outsiders, describes the process of adiabatic elimination of (fast) decaying variables. This technique is a common tool in some branches of physics[7, 24].

V

Concluding remarks

The immediate implication of the work discussed in this paper is to provide a tool for the analysis of Poincaré maps, allowing one to retrieve many relevant topological properties of the dynamical system contributing to the understanding of the topological organisation of the flow. The main result is to show that the braids associated with periodic orbits in 3-d flows can be determined up to a global torsion by inspecting the Poincaré map. This alternative to the (more intuitive) identification via the flow itself can be more efficient when analysing experimental data since the task of reconstructing the flow is usually harder than that of reconstructing the Poincaré map. Moreover, the present approach is at times the only alternative really available. Further, it is shown that the result still holds for strongly contractive higher-dimensional flows of the type discussed in Section IV. This result is particularly relevant for experimental physics, since the braid-orbit identification survives the fact that a considerable number of physically relevant 3-dimensional dynamical systems can in fact be regarded as reductions or truncations of higherdimensional systems.

VI

Acknowledgments

The authors are very grateful to Robert Gilmore (Dept. of Physics, Drexel U.), B. Gabriel Mindlin (Dept. of Physics, Universidad de Navarra) and Francesco Papoff (Dept. of Physics, Strathclyde U.) for encouragment and discussions. The authors are indebted to Ryszard Rubinsztein (Dept. of Mathematics, Uppsala University), Alberto Verjovsky (ICTP, Trieste), Moira Chas and Alicia Dickestein (Dept. of Mathematics, U. de Buenos Aires) for suggestions and challenging mathematicsoriented discussions. We thank the referee for directing our attention to T. Hall’s thesis and for his constructive criticism. We thank T. Hall for handing us a copy of his thesis during the authors’ stay at the Institute Non Lineaire de Nice, July 1993. One of us (HGS) aknowledges support from the “Fundación Antorcha” and a grant from Buenos Aires University. H. G. Solari is a member of the National Council for Scientific and Technological Research of Argentina.

Appendices A

Lemmas

We begin by giving the definition of fibration[26]: A map p : E → B is said to be a fibration if for every space X, every map fˆ : X → E and every homotopy, g : X × [0, 1] → B, with pf (x) = g(x, 0)

9

there is an homotopy gˆ : X × [0, 1] → E such that pˆ g = g. The situation is illustrated in the following commutative diagram fˆ

X

→

ω↓ X ×I

g ˆ

% g →

E ↓p B

where ω is the natural inclusion. In other words, if gˆ can be defined for all X (in a sense independently of X), we say that the triple (E, p, B) defines a fibration. In our problem E is the space (M, f0 ), p is the projection pn , B is the base space (Xn , x0 ), and x0 is a selected point, i.e. the one associated with a given periodic orbit or finite set of periodic orbits. Lemma 1 The triple ((M, f0 ), pn , (Xn , x0 )) is a fibration. Consider a given point q in X. Given fˆ and g, it follows that at t = 0 in I, the following relation holds: g(q, 0) = (g1 (q, 0), · · · , gn (q, 0)) = fˆ(θ1 ), · · · , fˆ(θn ) = pn (fˆ(q)) This holds for all q ∈ X. In other words, for each q, fˆ(q) gives a closed curve in R2 whose projection onto B coincides with g(q, 0). The function gˆ we are looking for can be constructed as follows: For varying t, the selected point in B (or equivalently the corresponding points in R2 ) move continuously as dictated by g(q, t). These selected points can always be connected by a simple closed curve in R2 ), i.e. they belong to the image of some ft ∈ M, and moreover, this curve can be deformed continuously as the points move, since E(≡ M) is path-connected (see Lemma 2 below). In other words, the ft ’s are homotopies starting in fˆ(q). This homotopy on E from t = 0 to t = 1 defines gˆ(q, ·): gˆ(q, t) = fqt (θi ) =

fqt ∈ M, where gi (q, t) = xi ∈ R2 ,

for some θ1 , · · · , θn ∈ S 1 . Neither fqt nor gˆ are unique. Lemma 2 (M, f0 ) is path-connected. We first note that if one restricts the elements of (M, f0 ) to be embeddings, this lemma is an immediate corollary of the Disc Theorem of Palais [27, pp 52], which proves that any two equioriented embeddings of the unit disk in Rk are ambient isotopic. For our purposes this translates into that there is a homotopy connecting any two elements of (M, f0 ). The restriction to embeddings would not alter the main result of the article. We produce however here a proof which is valid for C 0 functions. We use the discussion to introduce elements and ideas that will be used in the coming Lemmas. Proof: Consider an element f in M. Its image is a simple closed (Jordan) curve in R2 . Hence, one can always choose an arbitrary point z0 in the (simply connected) region interior to this curve. Regarding R2 as the Complex plane, the Riemann Mapping theorem [28, pp. 220-225] assures that there is a unique analytic function φ mapping the region interior to the curve defined by Im(f ) 1-to-1 to the interior of the unit disk, such that φ(z0 ) = 0 and φ0 (z0 ) > 0. Moreover, the inverse φ−1 of this function is also analytic. Since Im(f ) is a Jordan curve, φ−1 can be continuously extended to the border of the unit disk, |z| = 1[29, 30]. The extension is still 1-to-1. 10

Consider now the inverse image of circles of radius a, 0 < a ≤ 1 on the unit disk. By the 1-to-1 condition, they form a sequence of concentric curves moving from Im(f ) towards z0 . Analyticity and the fact that φ0 (z0 ) > 0, assure that for sufficiently small a > 0 these curves are arbitrarily close to a circle and thus trivially deformable into a circle of fixed radius. It remains to be shown that there is an homotopy connecting the restriction of φ−1 to the unit circle and f . After a suitable rotation or renaming of the θ coordinate in S 1 , we can make both functions coincide at the point θ0 = 0. It is easy to show that the composition s : S 1 → S 1 , i.e. s(θ) = φ(f (θ)) is homotopic to the identity: Consider the lifted exponential map sl of s. It is a continuous injective automorphism of [0, 1], mapping 0 to 0 and 1 to 1. The transformation F (x, t) = tx + (1 − t)sl (x) performs the homotopy. Thus, there is an homotopy sending any element in M to a circle of fixed radius. Hence any two elements in M are connected via homotopies. Lemma 3 π1 (M, f0 ) ' Z Proof: We divide the proof in three steps. 1. π1 (M) = π1 (M0 ): We call M0 the subset of M which has the point 0 in the interior of the region ˆ = M0 × R2 . It is clear that M ˆ and M0 have the same homotopy Ω bounded by f ∈ M and M ˆ → M, T (f, z) = f + z. The type[21, p.46]. Observe now that there is a surjective map T : M ˆ triple (M, T, M) is a fibration. The fiber is isomorphic to the Jordan domains Df defined by each f ∈ M. Then the result follows by exactness: ˆ → π1 (M) → π0 (Df ) = 1 1 = π1 (Df ) → π1 (M) To see that T defines a fibration, we note that (with the notation of Lemma 1) for every fˆ and g we can define a gˆ such that g(q, t) = T (ˆ g (q, t)). In fact, given the homotopy g(x, t) we can construct a homotopy gˆ(x, t) by choosing a point z(t) in the interior of each Jordan domain defined by f (t) (we call this interior Int(f (t)), somewhat loosely) in a continuous way with t. In such a case, the homotopy g(x, t) = (f (t) − z(t), z(t)) satisfies the fibration requirement since f (t) − z(t) ∈ M0 . It is not difficult to realise that it is possible to find a suitable continuous z(t) in many different ways, the only condition being that z(t) ∈ Int(f (t)) for all t. 2. π1 (M0 ) = π1 (S): We call S = {f : S 1 → S 1 , f continuous, injective, counterclockwise }, i.e. S is defined in the same spirit as M0 but where the image of the functions f are perfect circles. We want to show that paths in M0 can be continuously deformed to paths in S. To achieve this, we will again divide the proof in two smaller steps. We define first another auxiliary space, Mx = {f ∈ M0 : f = φ−1 }, i.e. more precisely those elements in M0 which can also be f |z|=1

regarded as the restriction to |z| = 1 of an inverse Riemann mapping from the unit disk to the 0 −1 domain defined by Im(f ), with the normalisation conditions φ−1 (0) = 0 and φ (0) > 0. f f The result follows now from the following facts: • There is an homeomorphism between M0 and Mx × S defined via f → φ−1 f [g] where g = φf [f ] ∈ S. • Mx is contractible. The first fact is immediate after checking that the elements in Mx vary continuously with f ∈ M0 (using the sup norm). Consider then any sequence fn → f in M0 . It induces the −1 (bounded and) normal family {φ−1 fn , φf } in Mx . Hence, by normality and Montel’s theorem, 11

we can pick any convergent subsequence out of the normal family, which will necessarily have the limit φf (see [30], pp. 316, for a similar discussion). The second fact can be proved by showing that there is an homotopy equivalence between Mx + + and the positive reals, given by the pair of functions F : Mx → R and G : R → Mx , where 0

(0) and the function G associates every positive real t to the function Gt , i.e. F (f ) = φ−1 f the Riemann Mapping tz defined on the unit disk |z| ≤ 1 regarded as an element of Mx . In other words, Gt : S 1 → R2 ; with Gt (θ) = teiθ . Note that we use throughout the dual nature of the elements of Mx i.e. that they are also Riemann Mappings and are thus defined within the whole unit disk and not only on the circle. It remains to be shown that both compositions of F and G are homotopic to the identities on the respective spaces. Clearly, F (G(t)) = t. The homotopy connecting f with G(F (f )) can be φ−1 (z)

performed with: H : Mx × [0, 1] → Mx where H(f, )(z) = f for 6= 0 and H(f, 0)(z) = 0 −1 φf (0)z. Continuity can be proved by standard methods using Cauchy’s estimate and Taylor theorem. 3. π1 (S) = Z: A loop in S is a family gt of circles, where the origin (gt (0) ∈ S 1 ) of each circle in the family loops continuously around S 1 . Hence π1 (S) is equivalent to the loops in S 1 . φ

Lemma 4 π1 (Xn , x0 )/p∗ (π1 (M)) → π0 (Yn (x0 )) induces a group structure on π0 (Yn (x0 )). Proof: We first observe that p∗ π1 (M) = Z(Bn ) with Z(Bn ) the center of the braid group (i.e. the group of global torsions). This property can be realised geometrically. In Lemma 3 we have shown that π1 (M) = π(S). The closed loops ft ∈ M (with t ∈ [0, 1] and f0 = f1 ) are topological tori fibrated by the (closed) orbits of θ. The projection p selects n of such orbits ft (θi ). Since the relative order among the images of θi has to be the same for all t being the elements in M injective functions of θ ∈ S the orbits ft (θi ) represent a global torsion in the pictorial presentation of braids. Using this observation toghether with Lemmas 2 and 3 the exact sequence reads Z ' π1 (M) → Bn → π0 (Yn ) → 1 or (after quotient out the global torsions) 1 → Bn /Z(Bn ) → π0 (Yn ) → 1 Hence, by the exactness of the sequence we have Bn /Z(Bn ) ' π0 (Yn ) which proofs the Lemma and concludes the proof of the Theorem 1.

B

Global Center Manifold Theorem for maps

In this appendix we consider the proof of the Global Center Manifold for maps in the form stated in Section IV (10). The proof follows the standard proof for the Center Manifold Theorem [23, 25, 31], though the necesary estimates are more intricate since the map for = 0 does not reduce to the identity.

12

As it is customary [23, 25, 31], we consider instead of the functions f, g in the map (10), the modified functions fˆ = f (xΨ(x), y, ) and gˆ = g(xΨ(x), y, ) where Ψ is a C ∞ function which is equal to 1 for |x| ≤ M and equal to zero for |x| ≥ M + 1. For convenience, we will scale y 7→ y so that the map (10) reads now: xn+1 yn+1

= fˆ(xn , yn , ) = gˆ(xn , yn , )

(12)

In what follows we also drop the ˆ to lighten the notation. We will search for the center manifold h in the Banach space E = {h : h ∈ C 1 , |h| ≤ M1 , |h(x) − h(x0 )| ≤ K|x − x0 |, |h0 (x) − h0 (x0 )| ≤ K 0 |x − x0 |} for some K, K 0 independent of h. We use sup norm for functions when needed. We divide the proof in the following steps: 1. The existence of h can be proved closely following the standard proofs [23, 25, 31] of the Center Manifold Theorem. Recalling that yn+1 = h(xn+1 ), we can write the functional equation for the center manifold as: −1 h(u) = g(φ−1 (13) h (u), h(φh (u)), ) where φh (x) = f (x, h(x), ). We can regard the right-hand-side as an operator O(h) and verify that it is a contraction operator acting on a complete Banach space. The estimation of the contraction rate of O(h) in the present formulation requires to prove the existence of φ−1 h and to establish its properties. 2. We show first that φh is invertible. Note that for sufficiently small , φh will be continuous and -close to φ0 . Consider next for fixed X, the map xn+1 = X − (φ−1 0 (φh (xn )) − xn ).

(14)

We claim that xn → xh (X), the unique fixed point of the sequence for each X. xh satisfies the −1 relation φ0 (X) = φh (xh ), i.e. letting Y = φ0 (X), we have xh (φ−1 0 (Y )) = φh (Y ). We need to prove that the iteration converges. Define ∆h (z) = φ−1 0 (φh (z)) − z. Then, since |xn+1 − xn | = |∆h (xn+1 ) − ∆h (xn )|, we need to show that |∆h (x0 ) − ∆h (x)| ≤ K|x − x0 |. Observe that φh (x) → φ0 (x) uniformly for all h ∈ E. Let −1 0 Qh (x, x0 , ) = φ−1 0 (φh (x )) − φ0 (φh (x)).

By the mean value theorem, we have, |∆h (x0 ) − ∆h (x)| = |Qh (x, x0 , ) − Qh (x, x0 , 0)| ≤ sup |D Qh (x, x0 , ξ) − D Qh (x, x0 , 0)|, ξ∈[0,]

where Dw denotes derivative with respect to w. Therefore, |∆h (x0 ) − ∆h (x)| ≤ 2 sup |D Qh (x, x0 , ξ)| = 2C|x − x0 |, ξ∈[0,]

where C = KDx φ−1 (2Kf Kh + KD f ) + M1 KDy f φ−1 (φh ) . 0

0

The K’s denote the corresponding Lipschitz constants. For sufficiently small the iteration converges and hence φ−1 h exists for all h ∈ E. 13

3. The previous reasoning shows also that φ−1 h is Lipschitz with respect to x. −1 0 4. Now we show that |φ−1 h (x) − φh0 (x)| ≤ C1 ||h − h ||. To this end we consider the limit points −1 of the iteration defining φh for the same X and any pair h, h0 : −1 |φ−1 h (Y ) − φh0 (Y )|

= |xh − xh0 | = |∆h (x) − ∆h0 (xh0 )| = |∆h0 (xh0 ) − ∆h0 (xh ) − (∆h (xh ) − ∆h0 (xh ))| ≤ 2C|xh0 − xh | + |∆h (xh ) − ∆h0 (xh )|.

The last term can be bounded as follows: |∆h (x) − ∆h0 (x)| ≤ Kφ−1 Kf |h(x) − h0 (x)|. 0

Collecting both estimates, −1 |φ−1 h (Y ) − φh0 (Y )| ≤

Kφ−1 Kf 0

1 − 2C

||h − h0 ||.

5. Finally, the existence proof is completed by noting that ||O(h) − O(h0 )|| = ≤

−1 −1 0 −1 ||g(φ−1 h , h(φh ), ) − g(φh0 , h (φh0 ), )|| −1 0 0 Kg (||φ−1 h − φh0 || + ||h − h ||) = C2 ||h − h ||

for some finite constant C2 and hence O(h) is a contraction operator on a Banach space for sufficiently small . The relevant properties of the space of C k -Lipschitz functions are discussed in [25, p. 41-43]. 6. Letting z = y − h(x) we have |zn+1 | = |g(xn , (h(xn ) + zn ), ) − g(xn , h(xn ), )| ≤ Kg |zn | = γ|zn |. Hence, the statements of the theorem follow for sufficiently small .

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[22] H.G. Solari, M.A. Natiello, B.G. Mindlin and R. Gilmore, Comments on the topological organization of 3-d flows and 2-d maps, Proceedings of the conference on Instabilities and Nonequilibrium Structures IV, Valparaiso, Chile, December 1991. E. Tirapegui and W. Zeller (Eds.) Kluwer Academic Publishers (in press). [23] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, 1981. [24] G.-L. Oppo, A. Politi, G. L. Lippi and F. T. Arecchi, Frequency Pushing in Lasers with with Injected Signal, Phys. Rev. A 34, 4000 (1986). [25] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976. [26] Sze-Tsen Hu, Homotopy Theory. Academic Press, N.Y. 1959. [27] A. Kosinski, Differential Manifolds. Academic Press, San Diego, 1993. [28] L. V. Ahlfors, Complex Analysis, 2nd. ed. Mc-Graw Hill, 1966. [29] G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol II. Wolters-Noordhoff, Groningen, 1969. [30] R. B. Burckel, An Introduction to Classical Complex Analysis, Vol. I. Academic Press, N.Y. 1979. [31] P. Hartman, Ordinary Differential Equations, Birkhäuser, 1982.

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fig1.ps

Figure 1: Braids

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fig2.ps

Figure 2: Braid product

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fig3.ps

Figure 3: Elementary braids

fig4.ps

Figure 4: Braid labelling

fig5.ps

Figure 5: Different braids but the same knot

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fig6.ps

Figure 6: Nails, strings and knots

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