Knotted periodic orbits in suspensions of smale's horseshoe: Period multiplying and cabled knots

Physica 21D (1986) 7-41 North-Holland, Amsterdam

K N O T r E D PERIODIC ORBITS IN SUSPENSIONS OF SMALE'S HORSESHOE: PERIOD MULTIPLYING AND CABLED KNOTS Philip HOLMES* Departments of Theoretical and Applied Mechanics and Mathematics and Center for Applied Mathematics, Cornell University, Ithaca, N Y 14853, USA Received 10 October 1985 Revised manuscript received 6 February 1986

Following earlier work on knotted periodic orbits in a suspension of Smale's [1] horseshoe diffeomorphism (Holmes and Williams [2]), we define a notion of iterated horseshoe knots. We show that approximately half the horseshoe knots are cabled and that, among these cablings, are infinitely many, noti-isotopic, iterated torus knots. We obtain uniqueness results for certain families of resonant torus knots, related to cascades of period-doubling and period-mfiltiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the H~non maps. We give formulae from which linking numbers can be computed and provide dosed form expressions for the type numbers of certain resonant iterated torus knots.

I. Introduction

In this paper we continue the study of knotted periodic orbits in three-dimensional dynamical systems begun by Birman and Williams [3] for the Lorenz [5] equations and generalized to flows with hyperbolic chain recurrent sets in [4]. For related work, see Williams [6], Bedient [7] and Franks and Williams [8, 9]. As in our previous paper (Holmes and Williams [2]), we concentrate on the "natural" suspension of the diffeomorphism of the two-dimensional disc known as Smale's horseshoe (Smale [1, 10]). In that paper we studied sequences of (p, q)-toms knots of period q (resonant torus knots) and proved existence and uniqueness results for pairs of such orbits. In conjunction with the kneading theory for one-dimensional maps and Hamiltonian bifurcation theory for area-preserving maps, these results permitted us to connect certain bifurcation sequences occurring as /~ increases for fixed e = 0 and e = 1 in suspensions of diffeomorphisms of R 2 of the form

(x,y)~F~,,~(x,y)=(y,-ex+f~(y)),

e ~ [0, 1].

(1.!)

Here f~: R ~ R has a single nondegenerate critical point and negative Schwarzian derivative. When (1.1) is the H6non map. See Holmes [11] for a summary of these results. In Holmes and Williams [2], we also obtained miscellaneous results on nonresonant torus knots and pretzel knots, and classified all knots up to period 8 for the horseshoe. It is well known that mappings like (1.1) undergo infinite sequences of bifurcations as # varies; the famous period-doubling sequence for the one-dimensional limit of the quadratic family

f~(y) =/1 _ y 2

y ~ # _y2,

(1.2)

*Partially supported by NSF CME 84-02069 and AFOSR 84-0051. The author wishes to thank Bob Williams for comments, suggestions and criticism.

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

8

Ph. Holmes / Knottedperiodic orbits in suspensions of Smale's horseshoe

discovered by Feigenbaum [12, 13], is only one example. Yorke and Alligood [14] have shown that countably many such period-doubling sequences occur during the creation of a horseshoe for rather general diffeomorphisms of R n. More generally, the occurrence of bifurcations to orbits of period q and rotation number p/q from an elliptic fixed point in the area-preserving limit of (1.1):

(x,y),---~(y,-x+f~,(y)),

(1.3)

(bifurcations leading to the resonant torus knots studied earlier) suggests that we study a broader notion of period multiplying bifurcations. This is the aim of the present paper. In the next section we review relevant sections of Holmes and Williams [2] and describe the template for the horseshoe and the associated kneading theory which permits us to order periodic orbits on the template and to describe their genealogies. The main results of this theory which we require are summarized in propositions 2.2 and 2.3, which deal with bifurcations, and 2.5, which deals with the factorization of kneading invariants. Also in this section we recall some ideas from knot theory. Readers familiar with this material can skip to section 3, where our main results begin. We define iterated horseshoe knots and the subtemplates on which they lie, and show how they can be characterized in terms of the *-factorization of kneading invariants. Proposition 3.1 describes this situation and fig. 6 gives an example. The remainder of the paper concerns special families of iterated horseshoe knots which arise naturally in period-doubling bifurcation sequences and 'resonant' bifurcations of elhptic periodic orbits of Hamiltonian systems. In section 4 we discuss period-doubling bifurcations and obtain formulae for linking numbers of the successive pairs of orbits created in sequences of such bifurcations. We show that all these orbits are iterated torus knots but that they are not generally algebraic knots. Theorem 4.3 and its corollary 4.5 are the main results of this section. We end with several examples and we compare earlier numerical studies of knotted orbits with our results. Section 5 contains results on sequences of cabled knots, which are created in period-multiplying bifurcations. These results can be seen as generalizations of the simpler notion of period-doubling cascades of dissipative maps and flows (cf. Yorke and Alligood [14]). In theorems 5.1-2 we state existence and uniqueness results for particular families of iterated torus knots and of cablings which need not be based on torus knots. Finally, in section 6 we indicate how the linking and genealogical information might be used in ordering bifurcation sequences in a manner similar to that of our earlier paper. Many of the ideas of this paper, including the crucial one of the subtemplate of section 3, were developed in conversations with Bob Williams. Other work on knotted orbits arising from period-doubling bifurcations has been done by Ueza and Aizawa [15], Beiersdorfer et al. [16], and Crawford and Omohundro [17]. We indicate connections between this work and ours where appropriate. For general background on dynamical systems, see Guckenheimer and Holmes [18] and for knot theory see Rolfsen [19].

2. The template and one-dimensional dynamics 2.1. The horseshoe template We start by outlining the procedure described by Birman and Williams [3, 4] for construction of a template. Let opt denote a flow in a three-manifold M 3 having a hyperbolic chain-recurrent set a2 (Bowen [20], Guckenheimer and Holmes [18]). Let - denote the equivalence relation z I - z 2 ¢~ ~p/(Zl) ---, opt(z2) as

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

9

7 ~

a

IC

b

Fig. 1. The horseshoe, its suspension and its template, a) The suspension of F, b) the template (,,~e"H, ~t).

t-", + oo. Applied to the orbits in 12, - collapses the flow along the strong stable manifolds, thus producing a semiflow g~t on a branched two-manifold .Xe'c M 3. The knot-holder or template (J{, ~,) has the property that any (finite) set of periodic orbits of % corresponds via isotopy to that of ~t: the knots and links induced on X" by ~, are isotopic to those of the original flow ~t on M 3. We illustrate the process for the horseshoe in fig. 1; the resulting template, (X'n, ~t) is the object of our study here. For more details on its derivation and relevance to nonlinear oscillations, see Holmes and Williams [2]. One can regard this 'reduction-by-one-dimension' procedure as providing a method complementary to the usual one of taking a two-dimensional cross section D c M 3 and defining an diffeomorphism F: D --, D; the Poincarb map (Guckenheimer and Holmes [18], chap. 1). In this connection, the equivalence relation can be viewed as follows. In the usual treatment of the horseshoe diffeomorphism, the chain-recurrent set 12-- N ,~_ _ ooF"(S) is described symbolically: to each point z we assign the bi-infinite sequence g'(z) = ( ~ }~=_ oo by the rule

xoa=

( x if FJ(z)~ Hx} yifFJ(z) He '

j

(2.1)

where Hx, HyCS are the strips indicated in fig. la. It is then proved that ~/': 1 2 ~ ( x , y ) z is a homeomorphism, and thus that f is topologically conjugate to the (full) shift o: { x, y }z ~ on the space of bi-infinite sequences of two symbols. See Bowen [20] or Guckenheimer and Holmes [18, chap. 5] for more information and generalizations. In this context, - corresponds to removal of the negative going portion of the sequences, since ~ ( Z l ) = ~(z2), Vj > 0 ,~ z 1 - z 2. This 'loss of history' is reflected in the noninvertible, one dimensional return map f induced by fit on the branch line I c .gffH: fig. 2a. 2.2.

Kneading theory

The symbolic dynamics outlined above implies that, to each finite acyclic word w with letters x, y, there corresponds a periodic orbit of % and hence a knot on (.xe"H, ~Pt)- Let ~ denote the infinite, periodic

10

Ph. Holmes/Knottedperiodic orbits in suspensions of Smale's horseshoe

i

~

=hi

l p. increosing

Ix

Ic

ly

a

I

b

Fig. 2. The induced one-dimensional map f,. a) The full shift f~, > hi; b) the bifurcating maps f,, /z ~ [sl, hl]. extension of w. If ~ ' = ok(F) for some k then w' and w correspond to the same knot. A collection of N, shift-inequivalent, finite, acyclic words corresponds to an N-component link. To compute the knot and link types it is necessary to order the words and their shifts so that the points in which the orbits intersect the branch I c .~e"H can be ordered correctly. The kneading theory of Milnor and Thurston [21] permits us to do this (cf. Guckenheimer [22, 23], Jonker [24], Guckenheimer and Holmes [18, chap. 6.3]), but before describing it, we generalize the class of maps f: I --, I to include t h e ' bifurcating' maps f~ of fig. 2b which are not surjective. This will enable us.to describe certain families of orbits in terms of their creation in the bifurcation sequences which the family f , undergoes as # increases. We assume there exist values s I < h 1 such that f~ has no non-wandering points for # < s 1 and a full shift for # > h r The theory outlined here works for C 3 maps with a single, non-degenerate critical point c ( f " ( c ) < 0, for definiteness) and the proofs are simplified if we assume that the Schwarzian derivative S ( f ) = f " " / f ' - 3 ( f , , / f , ) 2 is negative on I - ( c ) . Since we only wish to outline the theory necessary for the definition of period-multiplying sequences, we omit the exact description of the families f~, and other technical details which can be found in the references cited above, in Collet and Eckmann [25] or Holmes and Whitley [26]. The reader should think of the prototypical family z ~ # - z 2,

(2.2)

for which s 1 = - ~, h 1 = 2. For # > 2 this m a p has a nonwandering set ~ on which it is conjugate to a full two-shift on semi-infinite sequences of two symbols. Let z ~ I denote a point on the branch line or z E ~ = E1 ~_of-"(I) denote a point in the nonwandering set of the 'complete' two-shift. Let ~k(z)~ (x, y } z+ denote the semi-infinite sequence obtained by the rule

+j =

if i f ( z ) =

;

(2.3)

if f J ( z ) > ~p(z) is the itinerary of z. Let e(x) = + 1 , e(y) = - 1 and e(c) = 0, and define the inoariant coordinate of z

Ph. Holmes/ Knotted periodic orbits in suspensions of Smale's horseshoe

11

oo as O(z)= (0k}k_0, where

K

Ok(z) = l--[e(Lkj(z)).

(2.4)

j=O

Note that the sign of Ok(z ) detects whether f k preserves or reverses orientation near z (and if 0 k = 0 then ( f k ) , = 0). Let • denote lexicographical ordering on the invariant coordinate ( + 1 • 0 • - 1 ) . We have

Proposition 2.1. Let Z1, Z 2 E I (resp. ~ ) a n d z a < z 2. Then O(zt)•_ O(z2)(resp. O(zl)• 0(Z2) ). The following example illustrates how this allows us to compute knot and link information. Consider the words 'x2yxy ' and ' y ' . We list the shifts, the signed symbols and invariant coordinates below; writing ' + . . . . ' for ' + 1 , - 1 . . . . ', etc. Word and shifts

Signs

x2yxy

• ..

+

xyxvx

. . .

+ - - + - - +

yxvx

2

.

xyx2.F

• ..

),x2yx

.

y

.

.

.

+

--

+-q- _

.

.

.

"" • •

+

+

.

.

+

...

a 0

...

a t

...

+

+

.

+

. . . . +

.

Branch point

. . . .

++--

.

Invariant coordinate

.

.

.

.

.

. +

.

.

.

+

--

--

+

+ - - - - + +

.

.

.

.

.

--

+

+

+

...

a 2

_

_

+

•..

a 3

+

+

...

a 4

.

+

- +

. . . .

b

Alphabetizing the coordinates we obtain the ordering x 2yxy • x y x y x • x y x 2y • y • y x y x 2 • y x 2yx, or ao
I
< a 4,

and we can place the three points a0, al, a 3 on lx, since their words start with x, and the points b, a2, a 4 on Iy and connect a i to ai+ 1 (mod5) and b to itself with arcs to produce our link: fig. 3a. Note that we have to extend y to y - - y y y y . . . to correctly order it. Some elementary isotopy moves yield the braid presentation of fig. 3b, which shows that x2yxy is a trefoil linked by the trivial knot y. In fact any horseshoe knot can be arranged as a positive braid with braid axis ' y ' (Holmes and Williams [2, §6] and Franks and Williams [9]). The kneading theory does more than merely order periodic points; it also permits one to order bifurcation sequences for f~ and to describe decompositions of the nonwandering sets ~ of the non-surjective maps f~. In this respect the theory is prefigured in the 'U-sequences' of Metropolis et al. [27]. The orbit of the critical point is especially important. We define the kneading invariant as v ( # ) = lim_8(z).

(2.5)

z ---} c

since f~(c) lies to the fight of f~,(z) for any z(:~ c ) ~ I, 1,(#) provides information on which words w correspond to orbits realized in a particular non-surjective map f~. The monotonicity of the invariant

12

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

a

2YX~ Y b

Fig. 3. The link (x2yxy; y). a) On OCrH;b) as a braid.

coordinate (proposition 2.1) implies that for each z ~ I, O,(z) = 0 for all n, O(z) > ~,(/~) or O(z) < -v(t~). Using the fact that o(O(z)) = Oo(z)O(f~(z)) (cf. (2.4)), we see that for any n > 0 either Oh(z ) = 0 for all k > n or [o"(O(z))[ >_ l,(#), where 101 = 0 0 . 0 (i.e. if 00= - 1 , reverse all signs). A sequence a of the symbols _+1 is called v(#)-admissible if for all n > 0 a k = 0,

Vk > n

or Io"(a)l ~ ~(t~)-

(2.6)

Clearly O(z) for any z ~ I is v(#)-admissible, by construction. Conversely, if a is p(/~)-admissible, then there is a point z ~ I such that a = O(z), O(z-) or O(z+). Finally, any sequence fl which satisfies

Io"(/3)1 >/3,

Vn >_0

(2.7)

is called an admissible kneading invariant and can be realized as the kneading invariant of a map f , in the class considered here: i.e. there is some/~* ~ [sl, hi] such that p(/~*) =/3. A final ingredient in the result below is Singer's theorem [28], cf. Misiurewicz [29], which states that, for maps with a single critical point; negative Schwarzian derivative, and 'unstable endpoints', the critical point is asymptotic to a non-repelling periodic point, if one exists. In particular, this implies that for each /~ ~ [s 1, hi], f, has at most one attracting periodic orbit, and that the 'tail' of each admissible, periodic kneading invariant p(/~) corresponds to a periodic itinerary ~(#) with finite acyclic block w(#). To reconstruct w(/t) from p(/~), we reverse (2.4) to obtain

e(~b(fJ(c)))=uj(IX)/uj_l(l~);

e ( ~ k ( c ) ) = ~ , 0 ( # ) = +1,

(2.8)

and write x for + 1 and y for - 1. With the possible exception of the leading symbol, (2.8) shows that the resulting infinite word is periodic, since u(/t) is periodic. All admissible kneading invariants except the first (-Y-) begin with + . . . . , so that the itineraries begin xy . . . . Since f(c) is the rightmost point on the orbit of the critical point, f2(c) is the leftmost and thus truncation of the leading xy yields the lexicographically largest itinerary of this orbit. This itinerary is defined to be ~(#), and w(/~) denotes the finite acyclic word corresponding to one periodic block of ~(#). Note that if 7,(#) has period 2k then ~(/~) may have period k or 2k (see section 2.3). Proposition 2.3, below, provides information on kneading invariants and their associated words. These ideas lead to (cf. Guckenheimer and Holmes [18], theorem 6.3.4).

Proposition 2.2. There is an ordering for all finite, acyclic words wi of the letters x, y with the properties: (a) w1 and w2 are equivalent if and only if wl = o~(w2) for some k. (b) If f, is a one-parameter family of

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

13

maps such that the kneading invariants v(/1x), v(/12) are periodic and P(/11)l~ //(/12) , then in the interval [/11,/12] there is a set of bifurcations for f~ in which periodic orbits corresponding to each kneading invariant intermediate between ~,(/11) and v(/12 ) are created. For example, if v(#x)= + - + (periodically repeated) and /,t(/12)= " 4 - - then we infer the existence of a bifurcation value/1" ~ [/ix,/12] at which v(/1*) = + - - + - . The words corresponding to these sequences are xy, xy 2 and x y 4 respectively. Thus the period-5 orbit xy 4 is created after the period-2 orbit xy but before the period-3 orbit xy 2. T h e well-known ordering theorem of Sarkovskii [1964] (cf. Stefan [30], Li and Yorke [31]) can be proved using these ideas. In the simplest situation v(/1) decreases monotonically with increasing /1, leading to the universal 'U-sequence' of bifurcations exhibited by the quadratic family (2.2) (Metropolis et al. [27]) and a consequent monotonic increase in the number of periodic orbits. Finally we define the *-factorization of v(/1), which will be useful in discussing period-multiplying bifurcations (cf. Derrida et al. [32], Jonker and Rand [33]). Let X denote the set of admissible kneading O0 invariants. For a k-periodic v ~./ff and an admissible a ~vV" we define v * a = (%},=0 by %k+i = atvi;

0 < i < k.

(2.9)

Thus, if v = + . . . . . and a = + - + ..... we have k = 3 and v , a = + - - I + +1 + - - I + + I - + + " " " ; the vertical bars indicate the ends of (positive or negative) copies of the periodic unit of v. T h e *-process can be iterated indefinitely and the simplest such iteration leads to the Morse sequence, m. Recalling our notation of an overbar to denote periodic continuation, we have -

m=+*+-*+-*+-*...

=+--+1-++

I

++-I+--+1...;

i.e. the leading block of length 2 k is obtained by copying the leading 2 k-1 symbols and then repeating them with all signs inverted. This particular sequence describes the 'basic' period-doubling cascade which we treat in the next section. In fact, in general, if v is any admissible kneading invariant, then ~, * 4- - is the next lower admissible kneading invariant. This implies (via proposition 2.2) that every stable periodic orbit created for f as # increases must lose stability in a period-doubling bifurcation before/1 reaches h v We now develop this notion further. 2.3. Bifurcations o f f We recall specific details of the bifurcations of periodic orbits which the kneading theory predicts (Milnor and Thurston [21], Guckenheimer [23], Holmes and Whitley [26], als0 see Yorke and Alligood [14, 34]). Any periodic orbit V in the horseshoe suspension is one of two types: even or odd. Let z be a (k-) periodic (saddle-) point of F~ corresponding to such an orbit. In the former case the eigenvalues of DF~k(z) are positive, while in the latter they are negative. Yorke and Alligood [14, 34] call these orbits 'saddle' and 'MSbius' respectively, for in the latter case their local unstable manifolds are Mobius bands. In 'generic' families of diffeomorphisms leading to horseshoes, even orbits appear in saddle-node bifurcations, while odd orbits either appear in saddle-node or in period-doubling bifurcations. In terms of the kneading theory for f~, these cases are distinguished as follows. Throughout we assume that v(#) decreases monotonically as/1 increases, although this is not crucial.

14

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

C~~ (3

C

I

c

b a

a

c

b

b

Fig. 4. Bifurcations of f~. a) Case A; nk odd. b) Case B; n~ even.

Saddle-nodes. Let Pk be a k-periodic admissible kneading invariant which is not *-factorizable in the form ~,~,* + - for any admissible ~,~. Then when /~ reaches/z k at which 1,(#) first becomes ~'k, a saddle node bifurcation occurs in which a pair of k-periodic orbits Yk, Y~ are created. Initially both have the same itinerary, each periodically repeating block w~ of which contains an even number of y ' s , for f f is necessarily orientation preserving near the saddle-node, the eigenvalue ( f f ) ' being + 1. A s / t increases, a point on one of the orbits ('/k) crosses the critical point c (at/~ -- #'k), when ~,(/~) becomes ~'k * ÷ - and after which the two orbits ~ and 3'k are even and odd respectively and have the distinct itineraries which they will retain in the 'complete' horseshoe. There are two possible sub-cases: Case A: The itinerary { ~kj(c)}~= o of the critical point corresponding to ~'k contains an odd number n k of y ' s in the first k entries. In this case the point on Yk moves from right to left and each periodic block of the itinerary of Yk loses a y. Case B: { ~j(c))~= 0 corresponding to v k contains an even number n k of y ' s in the first k entries. In this case the point on "tk moves from left to right and each block of Yk gains a y. T o see this, we argue as follows. Let Jc -- [a, b] c I denote a small subinterval containing the critical point c, which is itself k-periodic for/t =/~'k. Then in case A, since an odd number of points on the orbit of c falls to the right of c, fkl[a, c)reverses orientation for /t near #~, while in case B an even number of points lie to the fight of c and fkl[a, c) preserves orientation. In other words, in case A we have f~(c) < f~(a), f~(b) and in case a f ~ ( c ) > f f ( a ) , f ~ ( b ) , see fig. 4. Finally, the words w~, w k corresponding to "y~ and "/k are derived from ~'k and ~k * + - respectively as described before proposition 2.2, above.

Period doubling: Let Pk = U~ * + -- be a *-factorizable 2k-periodic, admissible kneading invariant and let /~'k be the first parameter value at which v(/~) = u k. Then, by the arguments above, corresponding to ~'k we have a k-periodic orbit "Yk with word wk containing an odd number of y ' s . As # increases, p(/~) takes its next admissible value vk * + -" (at, s a y / ~ ' ); corresponding to which we have a 2k-periodic orbit "/2k with word w2k containing an odd number of y ' s . There is evidently an intermediate parameter value /~' ~ (/~'k,/~") at which "Yk undergoes a period-doubling bifurcation and "Y2k appears (initially with the same itinerary as ~'k). Again there are two subcases (fig. 4): Case A: The itinerary (~kj(e)}~= 0 corresponding to 1,k contains an odd number n k of y ' s in the first k entries. Then, as # passes #~", each periodic block W2k of "Y2kloses a y. Case B: ( qJj(e)}~_ 0 corresponding to Pk contains an even number n k of y ' s in the first 2k entries. Then as/~ p a s s e s / ~ " , each periodic block Wzk of 3'2k gains a y. We summarize:

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

15

Proposition 2.3(a). Let l,k be an admissible, k-periodic kneading invariant which does not factor as vk = v~ * + - for any admissible v~,. Let the itinerary { ~j(c)}~= 0 of the critical point c, corresponding to Vk, contain n k y ' s in the first k entries. Then, corresponding to ~'k * + - and vk respectively, there exist two k-periodic orbits "tk, ~'~ with finite acyclic words wk, w/,. Let yk, y~, denote the numbers of y ' s in w~,w~. Then y~ = n k + ½(1 - ( - 1 ) " 0 is even and we have Yk =Y~ + ( - 1) "~.

(2.10)

(b) Let v k = v~, * + - be an admissible 2k-periodic kneading invariant. Let { ~kj(c)}~°=o corresponding to v k contain n k y ' s in the first k entries. Then, corresponding to vk * + - and vk, there exist two periodic orbits ~2k, ~k with finite acyclic words w2k, wk of lengths 2k and k respectively. Letting Y2k, Yk denote the numbers of y ' s in W2k, Wk, we have Y2k = 2Yk + ( - 1)"k

(2.11)

and both Yk and Y2k are odd. All the words wk, w/,, w2k above correspond to horseshoe knots.

Remark 2.4. All possible (finite) words can be generated in this fashion. In the next section we will use the orbits ~ , ~'k (resp. "/k, ~'2k) to construCt a subtemplate A a ( J , k ) C ~ H which will carry the cablings of ~k- In this connection it is important to realize that the words w~, w~ correspond to isotopic knots K k, K[ (cf. Holmes and Williams [2, Theorem 8.2.5] and that wk, W2k correspond to knots K k, K2k which respectively form the center line and boundary of a M~Sbius band (cf. Yorke and Alligood [14, 34]. The following is easily verified: •

"p

1

Proposition 2.5. Let p~, v~; i = 0,1 be k,-periodic admissible kneading invariants. Let v = v°0* Vk~, 1' If Vko> 0 Vko, °J or if Vko=Vko o o' and v~, > v ~ then J, > v'. ~, - V k oO' , Vk. t

__

This proposition implies the ' b o x within a box' (boites emboltres) bifurcation structure discussed by o > G u m o w s k i and Mira [35], §2.7.2; [36], §1.2], see also Holmes and Whitley [26, pp. 55-57]. Since vk0 Vk 0o . + -- > " " " > v~00, m > • • • > pk00. + . . . . . . , the whole bifurcation sequence of f~ is repeated ' i n miniature' for f f0 restricted to a suitable subinterval. Every periodic orbit produced in such a sequence has a period which is a multiple of k 0, and the sequence must be exhausted before new orbits with periods not divisible b y k o can appear. This repetition of a gross behavior in miniature can be iterated indefinitely and is a basis for renormalization group studies of such maps. We use it in the next section to construct new families of iterated knots.

2.4. Cabling and iterated torus knots We end this section by reviewing some material from Rolfsen [19], Eisenbud and N e u m a n n [37] and Birman and Williams [3]. Let K o denote a knot in R 3 with solid torus neighborhood V0 and let l 0 c 0 V0 denote a preferred longitude: i.e. 10 bounds an orientable 2-manifold in the complement R 3 \ Vo. Let K 1 be a knot which lies on V0 and winds a times longitudinally and b times meridionaUy around Ko, (a, b) relatively prime, so that K 1 links K o b times (and crosses l o b times). Then K 1 is an a-cabling or a type (a, b)-cabling of K o.

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

16

If K o is the unknot, so that l o bounds a disc, then an (a, b)-cabling is an (a, b) torus knot. Let K 1 be an (al, bl) torus knot, and K, be an (a i, b~)-cabling of K i _ 1 for each 1 < i < r. Then K r is an iterated torus knot of type {(a i, b/)}~= 0 = ( ( a o , b 0 ) . . . . . (ar, b~,)). N o t e that each K i links K i 1 bi times. We call K 0 the core of K r. W h e n the period of K~ is l-[r=1 a~, we call it a resonant iterated torus knot. T h e adjusted type numbers (a~, b~) of an iterated torus knot are defined by b ' l = b 1,

b'i=bi-aiai_lbi_l,

2
(2.12)

and it is k n o w n that such a knot is algebraic if and only if a,, b; > 0

for each i = 1,2 . . . . . r.

(2.13)

F o r a definition of algebraic knots see Rolfsen [19] or Birman and Williams [3].

3. Iterated horseshoe knots We will n o w show how the *-factorization of kneading invariants, and the bifurcation structures e n c o d e d thereby, permit one to define families of iterated knots which provide a generalization of the iterated torus knots of section 2.4. Similar families could clearly be defined for other templates, such as that of Lorenz (Birman and Williams [3]). In Sections 4 and 5 we show that certain of these families are, in fact, iterated torus knots. Let k = (k o . . . . . kt) be a collection of l + 1 positive integers and _u= (u° °. . . . . u~,) be a collection of l + 1 admissible k n e a d i n g invariants having periods k 0 , . . . , k l respectively. Let N ( e ) = 1,Oo, uk, * " " " * u~,, be the k = Iq~=oki-periodic kneading invariant formed by the repeated *-product. By proposition 2.3, if u~, + - , then corresponding to N(_u) and N(e)* + - there are a pair of k-periodic itineraries with finite acyclic words w'(e ) and w(e) and a pair of corresponding periodic orbits 3,'(_u), 3,(_u), even and odd respectively. If utk,= + - , then corresponding to N(_~) we have a k / 2 - p e r i o d i c orbit 7"(_p). 7", ~" and 7 all c o r r e s p o n d to horseshoe knots K"(e), K'(e) and K(e). For l > 0, we call any such knot an iterated horseshoe knot with defining sequence N( u_). We n o w want to show that such an iterated horseshoe knot lies on a small ' s u b t e m p l a t e ' £~'(_u)c3ffH, which is d e t e r m i n e d by the sequence 1,°o* u~, * • • • * p/k,_1,. It is sufficient to establish this for p° ° and use induction. W e will show that, associated with v°o there exists a pair of knots K~, K o (father and mother) which respectively form a b o u n d a r y of and lie within the subtemplate (~,°o). T h e iterated horseshoe knots (the children) all lie within a solid torus Vo containing the m o t h e r K0, a b o u t w h o m they wind in a m a n n e r d e t e r m i n e d b y the last invariant ~,~,; the father K~ lies on the periphery 0 V0. Let w~ be the word corresponding to 1,0 o -_ Pgo/z 0 * + - for some admissible ~,°o/2 then w6 has ko" If Pko o + -. length k o / 2 , otherwise it has length k o. Let wo be the word of length k o corresponding to ~ko* D e n o t e the knots corresponding to w~ and w0 by K~ and K o and let them intersect the branch I c K H in the points b 0 . . . . . bk0_ 1 (resp. b o . . . . . bko/2 1) and a o . . . . . ako_ v S°(l,°o) cAe~H is a strip b o u n d e d on one side by K~ and containing K 0 in its interior. Since w~ (resp. wOwO) and w0 agree except in one letter, S°(1,°o) consists of k 0 - 1 ' s i m p l e ' strips connected subintervals of I x and Iy and joined end to end, together with one 'splitting' strip in which the loops of K~ and K o pass around opposite sides of ~ H F r o m the discussion before proposition 2.3, in case A K o goes to the left and K6 to the right, while the o p p o s i t e occurs in case B. See fig. 5. In this figure we also p e r f o r m a simple isotopy which reveals that

Ph. Holmes/Knottedperiodic orbits in suspensions of Smale's horseshoe

17

8

la

e

d Fig. 5. The subtemplate .La(V°o)c,Xe'H, a) Simple substrips. K6 can be on either edge. b) Case A, n, o odd. c) Case B, nko even. d) Flip right-hand (twisted strip to left). K6 can lie on either edge. Za(p°o) is merely a strangely embedded copy of O'~H, and therefore induces a map on its branch line, Jo, h o m e o m o r p h i c to f~ itself. This reflects the fact that ffolj ° repeats the bifurcation sequence of f~ll 'in miniature'; section 2.3. N o t e that, although we have moved the twisted strip of figs. 5b, c over to the left, the words of the knots on .£#(V°o) can still be read off correctly from their passage through the subintervals

Jx, Jv. T h e construction thus far has been geometrical. We must now show that ~a(po0) can be chosen so that

all knots K 1 having words w I corresponding to kneading invariants of the form ~,°o* u l lie within it, but without .~(~,°o) having self-intersections (other than on the branch line Jo)-

18

Ph. Holmes/Knottedperiodic orbits in suspensions of Smale's horseshoe

We use elementary properties of kneading invariants. Consider vk0 0 and Vko 0 , + ...... (repeated indefinitely), which are respectively the smallest and largest of all such v°0* v l . Consequently, all other words w1 corresponding to kneading invariants vko°, v~ must lie between w6 (corresponding to v°o) and w(~', corresponding to v°° * + . . . . . . on the branch line J1 =f~(Jo). (Recall that the word corresponding to a given periodic kneading invariant as constructed in section 2.2 is the left-most (largest invariant coordinate) of all shift equivalent words forming the periodic orbit 3%) We will show that fff(J1) (~ff(J1) = ~ for all j 4: k, thus implying that the intersections of .~8(V°o) with the branch line I are pairwise disjoint and hence that Za(V°o) can be chosen as described above. The kneading invariants have the forms Vko ... = 0 = + .... 1+ . . . . l + . . . . and v°0* + - - + .... I -- + "'" ] -- + "'" and so, from (2.4) and the rules for constructing words preceding proposition 2.2, the invariant coordinates of the itineraries ~ and ~ ' are ..... ....

+ I .... +-I""

+ I'",

(3.1)

+-I"'-

Note that 0(~6')I, 0(~6) so that w6 lies to the right of w~'. Now suppose that f / ( J 1 ) n f f ( J 1 ) 4 : ~ for some j 4= k. Since f~ is onto (for # > hi), this implies that ,11 nff°+k-J(Ja) 4: O and hence that an image of an endpoint, say a, of J1 lies in the interior of J r Thus 0(~')D, O(a), 0(~6) and the itinerary of a must agree with either w~' or w~ for the first k 0 symbols. But since a 4: ~ , ~ ' it must therefore have period > k 0, a contradiction. We summarize:

Proposition 3.1. Corresponding to any set _p= ( v~, }I=o of admissible k~-periodic kneading invariants, there exists a pair of knots K o and K~ (a mother and a father) and a subtemplate £~°(_v) which is a strangely embedded copy of .,UH and which carries a set of iterated horseshoe knots determined by _v (the children, grandchildren, etc.). Fig. 6 shows an example.

Remark 3.2. The parent knots Kr, K o with kneading invariants Vko, 0 v°° * vx, are examples respectively of companion and satellite knots (Rolfsen [19]). However, the template construction is considerably more

( Fig.

6. T h e s u b t e m p l a t e

~°(v)

f o r _u = + -

- - + • v~:

w o = x 2 y 3, wt~ = x 2 y x y .

Ph. Holmes/Knottedperiodic orbits in suspensions of Smale's horseshoe

19

restrictive than the general notion of companionship: for example, the satellite or child K t never 'doubles b a c k ' in the tubular neighborhood of Ko or K~. At the same time, the children include more general knots than cablings, although some of them are cablings, as we shall see.

4. Period doubling We now use the subtemplate construction of Section 3 in an iterative fashion to study some special classes of iterated horseshoe knots corresponding to particularly simple *-factorizations. We start with an elementary result to set the scene.

Proposition 4.1. Let wo be an acyclic word of length k o with an odd number Yo of y ' s . Let the horseshoe knot K o corresponding to wo have crossing number c o. Then among the horseshoe knots of period 2k 0 there is a 2-cabling K 1 of K o which links K o 2c o + Yo times. Corollary 4.2. Approximately half the horseshoe knots are cabled. Proof. Let V°o/2 * + - be the kneading invariant corresponding to w0. Then K 0 is the mother knot on the subtemplate .W(Vko/2 o ) and the desired cabling K 1 is the knot with word w I corresponding to Vko°, + _ , + _ . To see this, observe first that wI and WoWo differ in one letter, so that, on the splitting piece of .Z'(Vko/2 0 ) one strand of K 1 follows the father K6 (with word w~ corresponding to rko/2 o ) while the other follows the mother. In fact w 1 = WOWS,as the reader can check, so that K1 is a 2k0-periodic orbit, as required. The subtemplate construction also makes it clear that K 1 is the boundary of a M~Sbius band centered on K o and thus is a cabling: fig. 7. Finally, the number of crossings of K 1 over K 0 is 2c 0, inherited from the 'extrinsic' self-crossings of £a(r°o/2) due to its strange embedding, plus one crossing for each 'intrinsic' twist of •.~(1)ko/2 o ).. i.e. for each strand of K o on the right of Xe"n (in case B this includes the crossing in the splitting piece, cf. fig. 5c). Thus l ( K 1, K0) = 2c 0 +Y0, as claimed. It

~KO

KO K0

Ko

,oho,, twists s

_

__

twists

c Fig. 7. Intrinsic twisting of .£°(rk°o/2). a) Case A. b) Case B. c) MiSbious band.

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

20

W i t h a little more information on the kneading invariant corresponding to the ' m o t h e r ' knot K 0 we can go further:

Theorem 4.3. Let v ko ° = v0 * + - be an admissible, *-factorizable, 2k0-periodic kneading invariant corres p o n d i n g to a k0-periodic itinerary ~0 with an odd n u m b e r Y0 of y ' s in each finite acyclic block w0. Let the horseshoe knot K 0 corresponding to w0 have crossing n u m b e r c o and let the itinerary of the critical p o i n t c o r r e s p o n d i n g to vo have n o y ' s in the first k o entries. Then, for any l < oo, K o is the core of a sequence { K j }j= / 1 of k o • 2J-periodic 2-cablings with linking numbers satisfying

'(Kj+ Kj)=4J(2co+Yo)-(-1)n°(4'-(-1)j) 1,

(4.1)

5

a n d each such Kj is an iterated horseshoe knot with defining sequence j

factors

v k° o , + - - ,

... ,+--

R e m a r k 4.4. This is a ' k n o t and link' version of the period-doubling cascade results of Yorke and Alligood [14, 34]. See also Alligood [38].

Proof. This theorem is obtained by repeated application of proposition 4.1, f r o m the p r o o f of which we n o t e that the w o r d w 1 = WoW~ of K 1 has an o d d n u m b e r of y ' s , so that we can apply the proposition to K 1 to o b t a i n K2, etc. T o keep track of the linking numbers, we need the n u m b e r s of y ' s in the words at each stage: yj, a n d also the numbers of y ' s in the first k o • 2 j iterates of the critical point at each stage: nj. We first claim that, if n o is odd, then n I is even and vice versa. Using the definitions (2.4)-(2.5) and (2.8) we have for the two cases:

Case A: n o odd: ~°o:

itinerary of c

I+ .....

I-+"'+1

I+ ......

I+ ......

I

Wo .~

Y0 = no

no 0,-4---

Pk o

itinerary o f c

I+

.....

l+ ......

I - + ' " + l - + ' " + l +

I + ......

I .......

.....

l+ ......

W1

y l = 2yo + 1 n 1 = 2?10

I

[

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

21

Case B: n o even: 0

°

Pk o •

I+

....

itinerary of c

I+

......

+1-+

....

I I

I ...... w0

-..

yo=n0+l

j no

V0,+-k.

I+

....

+1-+...

I+

......

I .......

I

+ ....

I+

I+

......

I .......

....

+1

I

W1

Yl = 2 y o - 1 n 1 = 2n o + 1

Note that the kneading invariants and itineraries consist of copies of the initial segment with inversions in only the first entry of such segment. Applying this argument iteratively we obtain

3)= 2yj_l + ( - 1 ) "°+j, nj = 2nj_ 1 +

( 1 - ( - 1 ) "°+j) 2

(4.2) (4.3)

It now remains only to compute the linking numbers and crossing numbers of the 2-cabling Kj. Writing l( Kj+ 1, Kj) = lj+l, j and cj respectively for these numbers we claim that

lj+l,j= 2cj + yj,

(4.4)

cj+ 1 = 4cj + yj.

(4.5)

To verify (4.4)-(4.5), note that gj+ 1 is a 2-cabling of Kj which is isotopic to the boundary of a Mbbius band with Kj as center: thus at each crossing point of Kj, Kj+ x crosses itself four times and Kj twice, while each loop of Kj on the fight of X"H is crossed by one loop of Kj+x, which latter also crosses another loop of Kj+ x. See Fig. 8 for an example. We now solve (4.2) and (4.4)-(4.5) for lj+l, j to obtain (4.1) and complete the proof. The solution of (4.2) is

( a,n°(2J a,J)

(4.6)

as the reader can check by direct substitution and use of the 'initial' condition yj =Yo for j = 0. Next, we eliminate cj from (4.4)-(4.5) using

cj = 4%_ 1 + Yj-a = 2(2q-1 + Yj-x) - Y j - 1 = 21j,j-1 - Y j - 1 ,

(4.7)

Ph. Holmes/Knottedperiodic orbits in suspensionsof Smale's horseshoe

22

xy

Y

isotopy 3

isotopy_ xy3(xy) 2

Fig. 8. The first three period-doubling bifurcations. so that

lj+l,j = 2(2/j,j-1 - Y j - 1 ) +Yj = 4/j,j-1 + ( - 1 ) n ° + J ; J >- 1,

(4.8)

using (4.2). Finally we solve (4.8) subject to initial condition/a,o = 2Co + Yo from proposition 4.1, to obtain

~ =.1o ~ 1~o(4~ ~ 1~) Setting 11,o = 2Co +Y0 in (4.9) gives (4.1).

~49~ •

Ph. Holmes/Knottedperiodic orbits in suspensions of Smale's horseshoe

23

Applying theorem 4.3 to the 'basic' period-doubling cascade, we set J,°° = -~ * + - = + - + - + . . . . , so that n o = 0, Yo -- 1, and, since the base orbit K o is the unknot with word wo - - ' y ' , c o -- 0. Since ll, o = 2Co +Yo-- 1, K 1 is an iterated torus knot of type (1,1), (2,1) and so is also an unknot. In general, from (4.1) we have for j > 1

l(Kj+l'Kj)=4J-(-1)°( 4J-(-1)J) - 5

4J÷l-(-1)J÷15

(4.10)

The first nontrivial knot K 2 has period 4 and is a (2, 3)-torus or trefoil (Holmes and Williams [2]). Thus each Kj is an iterated torus knot with type numbers ay = 2 and by = (Kj, Ky_a)and adjusted type number b~ = by - a j a j _ l b j _ 1 = b j - 4 b j _ 1 = l j , j _ 1 - 4 l j _ l , j _ 2 = ( - 1) 0 + ( j - 1),

from (4.8). Thus we have

Corollary

4.5. F o r any 2 < l < oo, corresponding to the 2t+l-periodic kneading invariant + * + - • . . . * + - there exists a resonant iterated torus knot Kt of period 21 and type {(2, bj)}J= 2 with bj=(4J-(-1)J/5). The adjusted type numbers of such a knots are b ~ = ( - 1 ) j-1 and thus only K 3 = ((2, 3), (2,13)} is algebraic. This shows that the conjecture that all 'period-doubling' horseshoe knots are algebraic, made by Holmes and Williams [12, §8.1], is spectacularly wrong.

Examples T o conclude this section we compute some of the crossing and linking numbers for particular bifurcation sequences using the formulae and theorems proved above, and compare our computations with the numerical observations of Ueza and Aizawa [15] and Beiersdorfer et al. [16]. Also see Crawford and O m o h u n d r o [17, p. 177].

Example 1 The basic period-doubling cascade from ' y ' (n o = 0) Index: j

Crossing no.

Linking no.

No. of y ' s

Period

cj

lj, j_ 1

Yj

0 1 2 3 4 5 6 7

1 2 4 8 16 32 64 128

0 1 5 23 97 399 1,617 6,511

1 3 13 51 205 819 3,277

1 1 3 5 11 21 43 85

The entries in the last two columns should be compared with those in the penultimate row (.£a(x 0 + ~f2, Xo)) of table II of Uezu and Aizawa [15] and in the third column of table 1 of Beiersdorfer et al. [16].

24

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

W e suspect that Uezu and Aizawa's c o m p u t a t i o n of 203 rather than 205 = (4 5 - ( - 1)5)/5 is in error. See fig. 8 for the first few 2-cablings of y. Example

2

Period doubling from the (2, 5) resonant toms knot

Example

Index j

Period

Crossing no.

0 1 2 3 4 5

5 10 20 40 80 160

cj

x y 2 x y (n o =

Linking no. lj,/_ l

8 35 147 601 2,455 9.873

3) No. of y's

~)

19 77 307 1,229 4,963

3 7 13 27 53 107

3

H e r e we c o m p u t e that xy2xy 3 is one of a pair of pretzel knots b o r n with itineraries xy2(xy) 2 and thus we have case B with n 0 = 4. As c o m p u t e d in Holmes and Williams [2, §8], the genus of this knot is 5 and as a 7 strand braid (on JdH) it has 16 crossings. Here then lL0 = 2c 0 +Yo = 2.16 + 5 = 37 and we c o m p u t e the d a t a s h o w n in the table. Period doubling from the (7, 3, - 2) period 7 pretzel knot xy2xy 3 (cf. Holmes and Williams [2, §8]). Index j 0 1

2 3 4 5

Crossing no. Period 7 14 28 56 112 224

~)

Linking no. lj, j _ 1

16 69 285 1,159 4,673 18,767

No. of .v's Yl

5 37 147 589 2,355 9,421

9

19 37 75 149

A g r e e m e n t of the final column of the table above with the third column table 2 of Beiersdorfer et al. [16] is fortuitous here, since they are dealing with a period one orbit, presumably an u n k n o t (note that the u p p e r n u m b e r (4) in that table is incorrect). However, those authors give a general formula for the ' n u m b e r o f twists of the n t h ( = j - l s t ) (period-doubling) attractor with negative eigenvalues' (Beiersdorfer et al. [16], p. 272, also Crawford and O m o h u n d r o [17], p. 177.) A 'twist' corresponds to a y in the word, and so in the notation of this section, their formula becomes Y j = ((3Yb + 2)2 j + (-- 1 ) J ) / 3 ,

(4.11)

where Yb is the ' n u m b e r of twists of the first attractor in the period-doubling cascade when it has positive eigenvalues'. In our terms, this is the base orbit with index 0 and period k, hence n - 1 in their formula b e c o m e s j in ours. The formula (4.11) was apparently f o u n d by numerical investigation. Using the results of this section, we will show that the Beiersdorfer formula is correct for horseshoe 2-cablings o n l y for case B (n o even) orbits.

Ph. Holmes/Knotted periodic orbits in suspensions of Smale's horseshoe

25

From (4.6) we have

yj

=

(

2 J Y 0 - ( - 1 ) "° 2 J - ( - 1 ) 3

j

)

=2J(Yb+(_l)n0)_(_l),O

(2J 2,J) 3

where we note that in case A a y is lost as the base orbit moves over the critical point, so Yo =Yb in case B the reverse occurs and Y0 = Yb + 1 (cf. fig. 4). Rearranging (4.12), we obtain

y (j) = [(3y b + 2 ( - 1 ) " ° ) 2 , + ( - 1)"°+J]/3,

(4.12)

'

-- 1,

while

(4.13)

which agrees with (4.11) if and only if n o is even. Since embeddings of the horseshoe are prevalent in nonlinear oscillators and other three-dimensional dynamical systems (Guckenheimer and Holmes [18]), the general situation is evidently more complicated than Beiersdorfer et al.'s conjecture suggests. In this respect our work shows that, once a suspension has been chosen, the sequence of type numbers bj, along with the yj and crossing numbers cj, are completely determined by the word of the base orbit.Thus the 'chaotic renormalization orbits' conjectured by Crawford and Omohundro [17] are not realized in any specific flow. However, clearly by modifying the suspension of fig. 1, introducing additional twists, etc., one can produce infinitely many different sequences for any given base word. See section 6.

5. P e r i o d m u l t i p l y i n g

We start by reviewing the generic behavior of area-preserving diffeomorphisms in the neighborhood of an elliptic fixed point. This introduces the notion of period-multiplying sequences and provides a generalization of the doubling sequences of section 4. We also recall the construction of resonant torus knots of Holmes and Williams [2]. We then give the main results of this section, followed by their proofs and some examples. In this case the sequence of integers {(pj, qj)}~-0 defining the sequences are largely arbitrary (except that we require 0 < pJqj < 1 for each j), and so we cannot give closed form expressions for the type numbers of the resulting iterated torus knots. However, the formulae of theorems 5.1-5.2 allow one to iteratively compute type numbers for any specific knot.

5.1.

Area-preserving diffeomorphisms

The material in this section is taken from Meyer [39, 40], and Chenciner [41]; cf. Holmes and Williams [2] for a discussion in the context of the horseshoe suspension. The basic picture described here was certainly known to Poincar6 (cf. Arnold and Avez [42], p. 91 and Arnold [43], appendices 7, 8 and 9). Let z be an elliptic fixed point of an area-preserving diffeomorphism F~ : R 2 --* R 2 whose linearization DF~(z) has eigenvalues hi, 2 = e ± 2,~i0(~) 0(p) ~ (0, ½). The Hdnon map F~(x, y) = (y, - x +/~ - y 2 ) with /~ ~ ( - 1, 3) provides an example; z = ( - 1 + 1/1 + #, - 1 + V~-+ # ) in this case. For 'generic' F~ (those occupying an open-dense set in a suitable function space topology), certain nonlinear terms in the Taylor expansion of F,(z) are nonzero, we have a 'twist' map (Moser [44]), and the behavior near z is as sketched in fig. 9. The first point to note is that surrounding z there is a family C~ of smooth, invariant closed curves which contain orbits with irrational rotation numbers to. For our purposes it suffices to define the

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

26

~

~lenlorg e

Fig. 9. Dynamics near an elliptic fixed point.

rotation number of a point z as

p = p(F~, z) = limoo

( a r g ( F ~ ( z ) ) - arg (z)) 2-~n

}

"

(5.1)

The curves C~, form a Cantor set of positive Lesbesgue measure. These assertions form (part of) the celebrated Kolmogorov-Arnold-Moser (KAM) theorem. The second observation is that the gaps in the Cantor set are inhabited by alternating elliptic and hyperbolic periodic points with rational rotation numbers. In fact application of Poincart's geometric theorem (cf. Birkhoff [45]), together with generic hypotheses, shows that between any two KAM curves with rotation numbers wt < ¢.02,there are periodic orbits with all rational rotation numbers Po/qo ~ (Wl, °:2). One can apply the same reasoning to the elliptic members of these families and conclude that each such primary 'elliptic island' is itself surrounded by secondary elliptic islands of period qlqo. Continuing in this manner we expect to find k th order elliptic islands of period 1-Ij= k-10 qj for certain collections of integers qj. If one thinks of the suspension of such an area-preserving diffeomorphism, as in Holmes and Williams [2, §5], it is natural to conjecture that these islands correspond to iterated torus knots with type numbers aj = qj. In that paper it was shown (proposition 5.1) that the primary elliptic and hyperbolic points with rotation number p / q are (p, q) torus knots of period q for the 'natural' suspension of F~. Since the fixed point z eventually period doubles, at which point its eigenvalues are A1,2 = - 1, and since it originates in a saddle node bifurcation at which hi, 2 = + 1, there exist parameter values # ( p , q) such that the elliptic point of F~ has eigenvalues e ± 2~ip/q for each 0 < p / q < ½, terminating in e ± rri= 1. Thus in Holmes and Williams [2] we proved existence and uniqueness results for (p, q)-torus knots of period q for each 0 < p / q < ½ (theorem 6.1.2a). In this section we shall show that any sequence ((Pj, qj)}~=o of positive integers satisfying (pj, qj) = 1 and 0 < p j / q j < ½, defines a unique pair of iterated torus knots of period 1-[~_0 qj, and thus that in some sense all possible resonant bifurcations do occur during the creation of horseshoes. _

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

27

5.2. R e s o n a n t torus knots Before giving the main results we recall the construction of resonant torus knots from Holmes and Williams [2, §6], where it is shown that there are two and only two such knots K ( p , q), K ' ( p , q), odd and even respectively, for each (p, q) = 1 with 0 < p / q < ½. To construct K ' mark q points a o , . . . , aq_ 1 on the branch line I co~e"n in the natural order with q - 2 p on I x and 2p on Iy. Then write the integers 0 , . . . , q - 1 in the order 0 . . . . . q - p - 1, q - 1, q - 2 , . . . , q - p (invert the last p entries) and extend to a q-periodic sequence. Form a sequence of period q by writing 0, p , 2 p . . . . . m p . . . . using the order established above. Each block of this sequence begins 0, p and ends q - 2p, q - 1. Finally, lay an oriented circle on Ari_I so that it passes through the points a o . . . . . aq_ 1 in the order O, p . . . . . q - 2 p , q - 1 established above. To form K., move the leftmost point aq_2p o n Iy f r o m Iy to I x, where it becomes the rightmost point. Since the loop of K ' connecting aq_2p to aq_ 1 lies above all other loops it can also be lifted to the left. Note that the orbits thus created have rotation number p / q about the core orbit with word ' y '. The words w ( p , q), w ' ( p , q) and kneading invariant v ( p , q ) corresponding to the saddle node bifurcation in which this pair of knots is created have the forms: ~(p, q ) : + - - ( - ) [ + - ( - ) ] " "

+-(-)1 + .....

itinerary:xyx( x )[ yy( x )] . . . yy( x ) [yyx . . . wt: number of y's in word Y0= 2p number of y's in first q entries of itinerary n o = 2p - 1 ~(p,q)* + - : + - - ( - ) [ + - ( - ) l " "

+ - ( - ) 1 - ++ "'"

itinerary:xyx( x )[ yy( x )] . . . yy( x ) lxyx . . . w0:Y0 = 2p - 1 I-[ere ( - ) and ( x ) denote (possibly empty) strings of - l ' s and x ' s and ... denotes repetition of the unit in square braces. Note that w0 and w6 differ in the penultimate letter, w6 ending in ' y 2, i n w0 and ' x y ' . See fig. 10 for an example. Each acyclic block of v ( p , q ) contains p pairs of the form ' + - ' , while the remaining q - 2 p symbols are ' - ' . Finally we note that our construction yields a positive braid on q strands with crossing number c(p,q)

= (q-

2p).p +p.p +

1)

p(p2

1)

p(p+

2

=p(q-

1).

(5.2)

Here the first two terms come from the crossings of groups of q - 2p and p strands respectively (fig. 9(b)) and the second two from the half twists inthe p strand groups. The genus, g, of such a single component braid is given by the classical formula (of. Birman and Williams [3]): 2g = c - #st + 1

or here

(#st

2g=c-q+

(5.3)

= number of strands):

l,

(5.4)

28

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

ty

Fig. 10. The even resonant torus knot K'(2,7). To form K, move the strand connecting 3 to 6 from 1~. to Ix, which agrees with the formula 2g-- (p-

1 ) ( q - 1)

(5.5)

for a ( p , q) toms knot (Rolfsen [19]). 5.3. Iterated torus knots, crossing and linking formulae The main results of this section are:

Theorem 5.1. Fix l < ~ and an arbitrary sequence ~'t= ((pj, qj)}~=0 of relatively prime, positive integer pairs each satisfying 0 < p J q j < ½. Let l,qJj denote the kneading invariant corresponding to the even (pj, qj) resonant torus knot. Then corresponding to the kneading invariants N t = ~,°° • - . . • %t and N~ * + - , there exist iterated horseshoe knots K[, K t which are resonant iterated torus knots of type ((q0, P0), (ql, Poqoql - P x). . . . . (q,, b,)}, with the bj computable from the formula bj = qjqj_ lbi_ ~ + ( - 1)Jpj. Moreover, K~ t 0 qj with these type numbers. The adjusted type and K l are the only iterated toms knots of period q = 1-lj= numbers b~ of each such knot are ( - 1)Jpj and so none of them is algebraic for l > 0. Note that theorem 5.1 gives uniqueness as well as existence for resonant iterated torus knots. Existence will follow from iterative application of the next result, which generalizes proposition 4.1 and theorem 4.3, and uniqueness will follow from a slight generalization of the uniqueness proof for resonant toms knots in Holmes and Williams [2, §6].

Theorem 5.2. Let v = v°o*-T~- be an admissible, *-factorizable 2qo-periodic kneading invariant corresponding to a qo-periodic itinerary w0 with an odd number Yo of y ' s in each finite acyclic block wo. Let the itinerary of the critical point corresponding to v°o have n o Y's in the first q0 entries. Denote the horseshoe knot corresponding to wo by K o. Then for any pair of relatively prime, positive integers ( p l , ql) satisfying 0 < P l / q l < ½, there exists a type (qt, bl)'cabling K1 of K o among the horseshoe knots. K 1 has odd word

Ph. Holmes/Knottedperiodic

o r b i t s in s u s p e n s i o n s

of Smale's

horseshoe

29

w x c o r r e s p o n d i n g to the kneading invariant ~,°o* vq~ 1 * + - , where ~'ql ~ = ~'(ql, P l ) is the kneading invariant of the ( P l , ql) resonant torus knot. If P x / q l < ½ a second such cabling K { exists with even word w; c o r r e s p o n d i n g to ~,0. 1 If y~ and c i , i = 0,1 denote the n u m b e r of y ' s in the words w i of K~ and the qo J'q~" crossing n u m b e r s of K~ respectively, then the following formulae hold: Yl -- qlYo

+ ( -

bx--=ql(eo÷

1) "°(2pl - ql - 1),

~ ) + ( - ] ) n ° ( p x -

(5.6) ~),

(5.7)

(5.8)

cl=ql(qyCo+(ql-1)~)+(-1)'°(ql-1)(pl-~).

5.3. If Pl = 1, ql = 2, (5.6)-(5.8) reduce to the 2-cabling formulae of (4.2) and (4.4)-(4.5) o b t a i n e d earlier. Remark

T h e 'irregularity' of the iterative process using (5.6)-(5.8) with arbitrary p y , q j makes it difficult to give closed f o r m expressions for linking numbers in general, but there are nice special cases: Corollary 5.4. Let ~t of theorem 5.1 be {(p, q)}. Then K t (resp. where bj =p(q2j+2 period q1+1 with type numbers {(qj, bj)}j=o, t

5.5. Setting p = 1, q = 2 we recover c h a n g e d b y one). Remark

by =

is a resonant iterated torus knot of (-- 1 ) / + l ) / ( q 2 + 1), 0 < j < 1.

K/)

(4 j+l - ( - 1)J+1)//5, as in corollary 4.5 (with the index

W e first prove theorem 5.2 and then use it iteratively to obtain theorem 5.1 and the corollary. of theorem 5.2. As described in section 3, corresponding to 1,°° and v ° o * + - there are a pair of k n o t s K~, K o and the subtemplate . ~ ( p ° 0 ) c . ~ n which carries all iterated horseshoe knots with defining sequences of the form v°0* pql1. W e pick Vql = z,(pl, ql) to be the kneading invariant for a resonant torus k n o t of type ( P l , ql) and will show that the knots K { , K 1 corresponding to 1,°0.1,1 and V°o. ~,ql. + _ respectively are the isotopic ql-cablings of the theorem (if Pl = 1, qx = 2, the construction yields only one knot, K1). W e first tabulate certain features of the words W l , W { corresponding to these kneading invariants, using the relations (2.4)-(2.5) and (2.8):

Proof

C a s e A, n o o d d

~,~1

:+

(-)

~°o, ~,

:+ . . . . .

w;

:xy

~-)

~ - + . . . +)

(--}

.-.

(_+...+)+ ....

~-÷.-.÷)

~Oo,~,+-:+ .......

+ ... ÷ - + ... + ¢-+ ... +) + . . . . . .

• .. xy ... (.vy . . . ) + ... + ~-+ ... +) • .. + . . . . . .

xv ... (y.~.... ) x3. . . . +..,+(_+...+)_+...

wI

xy ""

xy ""

xy...

:xy

...

xy ...

""

yy ...

(yy ... )

yy ...

(yy ...)

xy ...

xv ...

xy ...

(yy "" )

• ..

xv...

()'5 ....

Y.~ . . . .

)

qo × ql

Vlth

V~o.~, w{

:

(-) (- + . . . . )

q-

:+ .... :xy

(-) !(- + . . . . )

(-) ... (- + . . . . ) + . . . . +

yy ""

v°o* vql • + - : + -

..4-+-..

W1

"• •

:xy

C a s e B. n o e v e n

yy " " "

+ x y •. •

-(-+

. . . . )+ . . . . + - +

( xy • • • )

yy • • •

.vy . •.

-(-+-'... ( x v • •. )

)

+•••

y) . . . .

+-+ y~. . . .

-(-+

.... )-+

(_,(~. . . . )

x.~. . . .

-

30

Ph. Holmes/Knotted periodic orbits in suspensions of Smale's horseshoe

I holf "~~,

Yo-~

j

~,

]

~.~J

a

b

Fig. 11. q l - c a b l i n g s of K o. a indicates the strand which can be m o v e d from J,. to J~ (resp. J , to J,. ) if ql > 2 P l . a) Case A; n o odd. b) C a s e B: n , even.

H e r e we use the fact that the qo'th entry of vqo ° is + 1 if n o is even and - 1 if n o is odd. We note that 0 with changes x --* y or y --, x only in the Wx(W{) consists of ql copies of the word wo corresponding to Vq0, p e n u l t i m a t e letter of each copy. W e will n o w give a geometrical construction of the cablings on ,Ea(v°0) and show that the associated w o r d s coincide with those of this table; since there is a one-to-one relationship between such finite acyclic w o r d s and knots, this will establish all but the formulae (5.6)-(5.8). It will be helpful to refer to figs. 5 and 11. W e first construct K 1 in case A (n o odd). M a r k Pl points d o . . . . . dpl_ 1 to the left of K o and Pl -- 1 points de1 . . . . . d 2 p , - 2 to its right on Jx and m a r k ql - 2 P l + 1 points d%l 1 dql_ 1 on Jy (left of K~). Similarly, m a r k q l - P l points e o . . . . . e q _ p _ 1 left of K 0 and Pl points eql_p~ , . . . . eql t right of K 0 on the b r a n c h Jo. Connect d o . . . . . dp _ 1 to e 0 . . . . . epl 1 and dpl, . . . . d2pl 2 tO e q l _ p l . . . . . eq _ 2 with strands on the untwisted piece of .W(Vq°o) and connect d2p ' . . . . . d q l _ 1 t o eql_pl l . . . . . epl with strands on the twisted piece of .Z'(v°0). Finally, c o n n e c t d 2 p l _ 1 t o the rightmost point eq~ 1. The remainder of £~°(V°o) is a r i b b o n e m b e d d e d in X"H with Y0 substrips on the right and so the ql branch points on Jo can be c o n n e c t e d in the obvious way with the ql branch points on Jx u Jy by strands which only cross when *W(V°o) twists or crosses itself. Since 2 p l < ql, this construction makes sense and if 2 p l < q], then the leftmost p o i n t d2p ' _ ~ can be m o v e d from Jy to Jx to yield K{. . . . . .

Ph. Holmes/ Knottedperiodic orbits in suspensions of Smale's horseshoe

31

We sketch this construction in fig. l l a , which also shows that K t and K[ are qx-cablings of K 0, since they can be arranged to lie on a (knotted) torus with K 0 as core. In case B (n o even) the construction is similar and we merely refer the reader to fig. l l b . We note that these constructions echo that of the resonant torus knots of section 5.2 (fig. 10). Let v t, v] denote the words of the knots K1, K[ of fig. 11, obtained by reading off x or y as K I ( K [ ) crosses the branch line of K n. We shall show that o I -- w1, V] ~ W~, where -- denotes shift equivalence (vl = ok(~x) for some k). We first observe that the construction of LP(V°o) shows that the itinerary v 1 agrees with wo (the itinerary of Ko) in all places except possibly those corresponding to points on Jx tA Jy (figs. 5-6). Thus, from the remark following the table, vx, v] agree with w1, w~ in all places except possibly (kq0_2) , k = 0 , 1 . . . . . qt - 1. We will only prove equivalence of Vl(V]) with Wl(W~) in case A (n o odd), since case B is similar and easier. We note that the Y0 - 1 (even) half twists and all the additional extrinsic crossings of LP(%°0) are irrelevant in determining the permutation of branch points on .Ix td Jy; we need only consider the splitting piece and the half twist immediately following it. From fig. l l a we see that, on this strip, the ql-strands of K t (or K [ ) suffer counterclockwise rotation through (qt - P l ) / q l , so that, adopting the numbering scheme PI . . . . . 2px - 1; Px - 1 . . . . . 0; qx - 1 . . . . . 2px for the branch points d o . . . . , d q _ l on J x U J y , these points are visited in the order 0, p t , 2 p x . . . . . m p t . . . . (mod ql)- But this precisely mirrors the construction of resonant torus knots in Holmes and Williams [2, §6.2] and section 5.2 above, except that the 2 p x - 1 points (resp. 2px points) d o . . . . . d2p~_ 2 (resp. d2p _1) lie in Jx and the remainder in Jy, rather than 2 p l - 1 (resp. 2 p l ) in Iy and the remainder in I x. Thus, starting the word with the branch point 0 = d2p 1_1 (strand a), the letters {( v t) kqo} ql k = 0 (resp. {( v t1) kqo} ~= 0 are precisely the inverse of those of the resonant torus knots K ( p t , q l ) (resp. K ' ( p l , ql)): k: 0

1

2...

vt:Y ""

x "'y

vi:x ""

x "" y "" (y "" )x "" x "" (y "")'"

""(y

"")x

"" x ""(y

"')'"

x "" x ""(y

""),

x "" x "" (y ...).

Now, consulting the table, we see that Vl - ° - 2 ( ~ ) and 6] - O-2(W~), as claimed. This also shows that K 1 and K [ each have a single component, since Pl and ql are relatively prime. In fact the (clockwise) rotation number relative to the core K 0 is - ( q l - P x ) / q l = P l / q l (rood 1), as the area preserving theory of section 5.1 requires. We remark that in case B, the rotation on the splitting strip is P t / q t counterclockwise and, as figs. 10 and l l b show, K 1 and K~ behave on the splitting piece of LP(V°o) precisely as do the resonant torus knots on o~/"n. Thus one obtains the words Vl: X " ' ' y

"'" X " ' ' ( X " ' ' ) y " ' ' y " ' ' ( X " ' ' ) ' ' ' y

vi y...y...x...(x...)y...y...(x...)...y...y...(x...)=w

"''y "''(X "'')=

W1 2

as required. The constructions above show that, in case A, the number of y ' s in w1 is qlYo + ( q t - 2pl + 1), since w1 consists of ql copies of wo with qt - 2P1 + 1 x ' s replaced by y ' s (there are ql - 2Pl + 1 branch points on J,,). Similarly in case B q l - 2pl + 1 of the copies of w0 in w1 have y ' s replaced by x ' s and we have

32

Ph. Holmes/Knottedperiodicorbitsin suspensionsofSmale'shorseshoe

Yl = qlYo - (ql - 2p~ + 1). Thus we obtain formula (5.6). Also note that Yl is odd and thus that the words w~ are even, since they contain one less (resp. one more) y than w v T h e formulae for crossing and linking numbers can be c o m p u t e d by inspection of fig. 11. We indicate the arguments in case A. K 1 does not cross K o between Jx U Jy and Jo, but in the following half twist, ql - P l strands of K 1 cross K 0. This is followed by (Yo - 1 ) / 2 full twists, in each of which all ql strands of K 1 cross K o. In addition, the c o 'extrinsic' crossings of £f(%°o), induced by those of K o, lead to qlCo crossings. Thus we have

I ( K 1 , K o ) = qlco + ql

+ ql - P l ,

or

bl=ql(Co+ ~)-(P1-q1/2), as in (5.7) with n o odd. Similarly, since each half twist in a ribbon with k strands causes k ( k - 1 ) / 2 self-crossings, K 1 has (ql - 2 p l ) ( q l - 2 p l - 1 ) / 2 +Pl(qX -- 2 p l ) self-crossings between Jx to Jy and Jo a n d a further ql(ql - 1 ) / 2 "Y0 as a result of the Yo half twists of &o(V°o). The c o extrinsic crossings give a further q2Co, leading to

Cl=q21CO+q1(q1--l)~f +(ql--2p1)( pl+

q1--2pl--l))2

=ql(qlCO+(ql--1)~)--(ql--1)(pl--~), as in (5.8) with n o odd. In case B the linking and crossing numbers can also be read off from fig. 11 and one obtains

:ql o+qi( ) =ql(Co+~)+(Pl--ql/2),

clq= co+q,,ql -

+2

(p Ipl ,) 2

+Pl(Pl+ql-2pa))

=ql(qlCO+(ql--1)~)+(ql--1)(Pl--ql/2), in agreement with (5.7)-(5.8) with n o even. This concludes the p r o o f of theorem 5.2.

•

Proof of theorem 5.1. We apply theorem 5.2 successively to obtain Kt, K/. We first deal with (a) existence. H e r e the core K o is itself a torus knot and thus, from the discussion at the end of section 5.1, n o = 2po - 1 is o d d a n d Yo = 2po - 1. Also the crossing n u m b e r c o of K o is c o = P o ( q o - 1). Applying theorem 5.2 we construct a ql-cabling g 1 of K o which is an iterated torus knot of period qoql and type (qo, Po), (ql, bl)

Ph. Holmes/Knottedperiodic orbits in suspensions of Srnale's horseshoe

33

where 2po- 1

(5.9)

bl=ql(Po(qo-1)+---'~)+(-1)2P°-l(pl--~)=poqoql-Pl, from (5.7). Since K 1 has Yl = q , ( 2 p o -

1) + (-- 1)2p°-1(2pl -- ql -- 1) = 2(poq I - - P l ) + 1,

(5.10)

y ' s in each word Wl, we can apply theorem 5.2 again to produce a q2-cabling of K 1 iterated torus knot of period qoqlq2. However, to compute Y2 and b2 we require nl: the the first qoq~ entries of the itinerary of the critical point corresponding to the kneading i = v(pi, q~) are kneading invariants of resonant torus knots, we use Since b o t h the 1,Pqi section 5.2 to write:

v~

+

ql

.....

°+

o.

'

itinerary

-

_

-÷...?-÷.

÷_

+

(-)

(-,

_ _ ii. ÷

(-

+

.

.

.

-

.

.

+ )) ÷ + . .. ..... . .

-

.

÷

+-

+

.

.

which will be an number of y ' s in invariant v°° * vql" 1 the discussion of

(-)

-

.

.÷i •÷ . -~ - -

.-

- ,~÷- - i ; . ÷ (( -.....÷ " + )

)

We see that the first qoql entries of the itinerary for v°o* VXqlis ql copies of that for v°qo with the leading ' x ' replaced by a ' y ' in ql - 2Pl of those copies. Therefore n I = noq 1 + (ql - 2 P l ) = (no + 1)ql - 2Px is even, since n o is odd. N o t e that each periodic block of v°, * vlq~ ends in + 1, so that, applying the * -factorization once more, we have v2

+

q2

I)qo * Pqi * q2 o

itinerary

+

-

_

(-)

.... +__\ ....• \\.i:._,_+( +

-

+

.... ,7.:..+:+

....

......

+

.

.

.

.

(+ -

.

.... iii

T h e n u m b e r of y ' s in the first qoqlq2 entries of the itinerary is q2nx + (2p2 - 1), since we have q2 copies of the itinerary for vOo, vlq~ with 2p2 - 1 leading 'x's' changed to 'y's'. Thus n 2 is odd and v°qo* vql, vEq2ends 0 Continuing indefinitely, we see that nj is odd for j even and vice versa, so that in - 1 , as did Vqo" ( - 1) n, = ( - 1) j+l and we can write (5.6)-(5.8) as iterative formulae for any j >__1

Yj = qjYj-1 + ( - 1)J(2pj - q j - 1),

(5.11)

bj=qj(cj-l+~-L)+(-1)J(Pj-~),

(5.12)

cj=qj(qjcj_l + (qj-1)~-L)+

(-1)J(qj

- 1)( p ) - ~-~J).

(5.13)

l Application of theorem 5.2 can continue indefinitely to produce a period q = FIj= 0 qj iterated torus knot K / a n d its partner K/, as claimed. It merely remains to eliminate yj and cj from (5.11)-(5.13) and obtain a recurrence relation for the type numbers b r

Ph. Holmes/ Knottedperiodicorbitsin suspensionsof Smale'shorseshoe

34

From (5.13) we have C,/ 1 =

qj-12)]--qj

q.l-l[qJ X(¢j -2-Jc~2"~)+(-I)j-I(pj-I

=qs-lbj-l-qS

-1

2

(-1) j-

,(

Yj 2 1---2

(_l)j

qJ2 ,) "

Pj 1

l ( p j _ l -- "--~)qs-1

(5.14)

Substituting (5.14) into (5.12) we obtain

bj=qjqj-lbj

I+

qJ

~(Yj-I--qj-lYj

2)+(--1)

j

(

qs ( " P j - y + q / Pj-1-- -qj-1))2 -

(5.15)

F r o m (5.11) we have

Yj , - q j - , Y j - 2 = ( - 1 ) J - l ( 2 P s - , - qj-1 - 1), which we use in (5.15) to obtain

b/= qj qj_l bj x + ( - 1) Jpj,

(5.16)

as claimed. Thus, from (2.12) and (5.16), theadjusted type numbers are

b; = b i - qjqj_ xb)_ l = ( - 1 ) J p j . (b) Uniqueness. In Holmes and Williams [2, theorem 6.1.2a] each pair of resonant (p, q) toms knots with ( p, q) = 1 and 0 < p / q < ½ was proven to be unique by arranging the knots as positive braids on p strands and showing that, among all such p-strand horseshoe braids of period q, they alone maximise the crossing number and hence, via (5.3), the genus, g. That g is an isotopy invariant is well known (Rolfsen [19]); q is an invariant for the particular template and p is an invariant in view of: Theorem 5.5. (H. Morton). If B is a positive braid on p strands which contains (i) a full twist and (ii) in addition each elementary braid crossing oi, i = 1 . . . . ,p - 1, then p is an invariant of B. In fact p is the positive braid number of B: the smallest k such that B is isotopic to a positive braid on k strands, see Franks and Williams [9]. The proof of theorem 6.1.2 of Holmes and Williams [2] is complicated, since the template ~t"~H has first to be rearranged in its braid form to produce the required twist and elementary crossings. Here things are simpler, but we will use a similar argument inductively. First observe that, at any stage j, candidates for resonant iterated torus knots of type ( ( q i , bi)}J=0 must have period I-If= 0 qi and defining sequences of the form V°*qo "'" * VqJj' where va= defY°o* . . . * VqJ~ defines a ((qi, bi)}{ - d resonant iterated toms knot. Moreover the candidates must lie on the subtemplate .~e(va). Suppose then that the pair of iterated torus knots K}_ x, Kj x corresponding to va, v~ * + - and satisfying b i = qiqi lbi 1 "}- ( -- 1)ipi; 1 ~_~ i < j; b o =P0, are unique. The argument starts with uniqueness of the initial pair of resonant torus knots with defining sequence /)a = POo = P(Po, q0) and, since the construction shows that only Kj_ x can be cabled by knots with defining sequences of the form /Pa* P~,, if we can prove that va, qj and pj uniquely specify two resonant

Ph. Holmes//Knotted periodic orbits in suspensions of Smale's horseshoe

35

Ko tl

B

r

-y-~ /z

do

/.c

a

/,i

b

Fig. 12. General iterated torus knots on -~a(r,a). a, fl, "f, 8, p., l, represent n u m b e r s of strands in each ribbon, a) Case A: j odd. b) Case B: j even.

iterated torus knots Ks, Kj, then induction can proceed. Note that, via the formulae (5.11)-(5.13) and (5.16), specification of {(qi, Pi)}/-0 is equivalent to specification of {(q~, b,)}{= 0. Since the subtemplate Sa(va) has at least y o / 2 = ( 2 p 0 - 1)/2 'intrinsic' full twists and c o = P 0 ( q 0 - 1) self-crossings, the braid formed by any knot lying on LP(va) c.gg"H satisfies the hypotheses of theorem 5.5 after rearranging K j _ 1 and Ks_ 1 (and thus £a(va)) as braids on Po" qx . . . . qj-1 strands. If P0 > 2 then we have 1½ full twists automatically, while if Po = 1, then qo > 2p0 = 2 and we have ½ full twist and at least 1 self-crossing, which can be pulled out to give a further full twist: cf. Holmes and Williams [2], fig. 12d and see fig. 13c.) Thus the number Po" qx . . . . qj is an isotopy invariant, along with the period qo" qx . . . . q j, for any iterated horseshoe knot with defining sequence va * /-PqJffSince the genus gj is a knot invariant, we see that, via the formula (5.3),

2 g j = cj -

J [-[ qi + 1,

(5.17)

i=0

the crossing number cj is a third invariant. We shall show that, among all the iterated horseshoe knots of (positive) braid number Po" ql . . . . qj and period qo" ql . . . . qj which link K j _ t bj = qjqj_xbj_~ + ( - 1)Jpj times, only K j and K s extremise the genus and hence the crossing number cj. In case A ( j and n j_ t odd) the extremum will be a minimum; in case B ( j and nj_ 1 even) it will be a m a x i m u m . From the proof of theorem 5.2 we note that most of the contribution to linking and crossing numbers of candidate I-I,S=o q;-periodic, iterated horseshoe knots is determined by the y j _ , - 1 intrinsic half twists and cj_~ extrinsic crossings of S°(Va) (fig. 11). The splitting segment (and in case A the half twist immediately following) are the only places where variations determined by VqSjcan occur among the knots with defining sequences Pa * VqJj• Consider then the general arrangements of a + fl + "/= qj strands of fig. 12. Since this section of Sa(va) is followed by an even (Yj-1 - 1)/2 number of full twists and the braid is connected top to bottom in the obvious order, we see that CaseA:

v=a,

#=fl+~,;

CaseB:

/z=a+fl,

v='t.

(5.18a, b)

Ph. Holmes/Knotted periodic orbits in suspensions of Smale's horseshoe

36

a

b Fig. 13. Examples of cablings and a non-cabling, a) x2y and x2yxy2(x2y)2xy2; b) x2y 3 and x2ya(x2yxy) 2. c) x2y and

X2)'( X.V2)2( x2y ) 2.

The linking numbers

I(K1,Ko) of the appropriate iterated torus knots are (eq. (5.12))

bj=q/(c/_l + ~ - l + (-1)J(pi-qJ2) =q/(c/_ 1

-

I + Y~J - I - 1 )

qJ +-~+(-1)J(Pj-qJ2),

(5.19)

so that the linking contribution of the segments of fig. 12 are CaseA:

qj-p/~l~=qj-pj;

CaseB:

pj~B+8=&.

(5.20a, b)

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

37

Isotopy

'component on 6 strands

)c~ H

5 ~ twists

Fig. 13. (continued).

qj and ot=pj

F r o m a + fl + y = CaseA:

(5.18a, b), (5.20a, b) we obtain CaseB:

ot=qj-2pj+6,

#=pj-8

#=pj-8

7=qj- 2pj+8;

Y=pj

(5.21a, b)

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

38

Here 8, which specifies how the strands 7 and a respectively split, remains as a free (non-negative) parameter. We now compute crossing numbers from pj, qj and & The only indeterminacy occurs at the branch line Ja and this is removed by seeking extrema. In case A we minimize cj by requiring that no unnecessary crossings occur, to obtain

(

cj = qi qjcj_ 1 + (qj-

1)

]2-

=qj(qjcj a + ( q j - 1 ) ~

+

q j ( q j - 1) ~ ( ~ - 1) 2 + 2 + f l ( ' y - 8)

-~)-(qj-1)(pj-

~)+

~(8-1).

(5.22)

This is greater than the crossing number (5.13, j = odd) unless 8 = 0 or 1, corresponding to the resonant iterated torus knots K / a n d Kj respectively, when equality obtains. Thus Kf and Kj represent minima of the crossing number and so are unique. In case B we maximize cj by introducing as many crossings at the branch Ja as possible:

cj qj(qjcj_l+(qj_l(YJ-l-1)) =

2

= qj(qjcj_l+ (qj-1)~-Z)+

+

(fl + ~,)(fl + 3, - 1) 2

(qj-1)(pj-

+BS+(a-8+8)v

~---~J)- 26--(8- 1).

(5.23)

This is less than (5.13, j = even) unless 8 = 0 or 1, when we have equality and obtain the resonant torus knots K} and Kj respectively. Thus K / and Kj maximise the crossing number and so are again unique. In both cases we have uniqueness and induction can proceed. The proof of theorem 5.1 is complete. •

Proof of corollary 5.4. qj = q, Vj and subject

As in the proof of theorem 4.3, we solve the recurrence relation (5.16) with pj = p, to initial condition b1 =Poqoqx -P~ =p( q2 1). This yields

bj=p( q2j+2- (-1) j+l ) ~5 + 1

as the reader can easily check.

5.4.

'

(5.24) •

Examples

We give three examples of period 15 iterated horseshoe knots, one of which is a resonant torus knot of the type dealt with in theorem 5.1, one of which is an 'irregular', non-resonant, iterated torus knot, with a (2, 3) non-resonant toms knot of period 5 as core, and the last one of which is not a cabling.

Example

1. Here we pick (p0, q o ) = ( 1 , 3 ) and ( p l , q l ) = ( 2 , 5 ) . The kneading invariant vq0 ° * + - of theorem 5.2 is + - - * + - = + - - - + + , with associated word w0= x2y, and the kneading invariant of the (2, 5) torus knot is + - - + - . The words %, w~ of the 5-cablings K 1, K ; of K 0 can be obtained

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

39

either f r o m the geometric construction of the p r o o f of theorem 5.2, or by writing the kneading invariants gl:

+ - - . + - - + - . + -

K[:

+--*+--+-

=~wl=x2yxy2(x2y)2xy2,

=w[=x2yxy2(x2y) 3,

a n d c o m p u t i n g their associated 15-periodic itineraries w0, w~ and words as described in section 2.2. N o t e that n o =Y0 = 1 and Yl = qjYo + ( - 1)n°(2Pl - ql - 1) = 5.1 + ( - 1)1(2.2 - 5 - 1) = 7. Lexicographical ordering of the invariant coordinates of K 0 ( x 2 y ) and g 1 yields fig. 13a. g I (and K [ ) is a (3,1), (5,13) iterated torus knot, or, since x2y is trivial, simply a (5,13) (non-resonant) torus knot of period 15.

Example 2. Here we pick 1,°o-- + - - - + , corresponding to the core knot K o with w o r d x 2y 3. As shown in H o l m e s a n d Williams [2, §8], this is a (non-resonant) (2, 3) torus knot of period 5. Here n 0 = 2 and we have case B of theorem 5.2. Setting P I -- 1, ql = 3 we construct the 3-cabling K 1 of K 0 shown in fig. 13b with w o r d w 1 = x2y3(x2yxy) 2, for which Yl = qlYo + ( - 1)n°(2Pl - ql - 1) = 3.3 + ( - 1)2(2.1 - 3 - 1) = 7. N o t e that w I is the word associated with the k n e a d i n g invariant + - - - + * + - - * + - and that K t is a (3, 2), (3, 22) iterated torus knot and so is an algebraic knot with adjusted type n u m b e r b I = 4.

Example 3. H e r e we pick v°° = + - - so that K 0 has word x2y as in Example 1, and we construct a period 15 k n o t K 1 using the period 5 kneading invariant + - - - + . The associated word corresponding to + - - * + - - - + * + - is w I = x2y(xy2)2(x2y) 2. Lexicographical ordering of w I and w0 and their shifts yields figure 13c. The linking n u m b e r 11,0 = 13, as in example 1, but the crossing n u m b e r cl of K 1 (as a positive braid on 5 strands) is 5 . 5 ( 5 - 1 ) / 2 + 4 = 54. This should be contrasted with the crossing n u m b e r of the first example (again as a 5-braid): 5 . 5 ( 5 - 1 ) / 2 + 2 = 52, thus providing an illustration of the uniqueness p r o o f of theorem 5.1. In this example K 1 is not a cabling.

Remark 5.6. I n view of the fact that, a m o n g all horseshoe knots with positive braid n u m b e r p and period q, the ( p , q) resonant torus knot(s) maximise the genus at g = ( p - 1)(q - 1 ) / 2 , we can give upper b o u n d s for the genera and crossing numbers cy of resonant iterated torus knots as q = I-If_ 0 qi-strand braids d e t e r m i n e d b y ~1 = {(Pi, q~)}~-o (eq. (5.13)). As shown in the p r o o f of theorem 5.1, such a knot has positive braid n u m b e r p = Poql... q and period q = qoql.., q and we therefore obtain via (5.3) and (5.5)

2

, cj
' I-Iq,"

i=1

)

q,-1 .

(5.25)

F o r example, the (5,2),(7,68) iterated torus knot determined by {(2,5),(2,7)} ( v = + - + - - * +--+--) has crossing n u m b e r 464 (and genus 215) c o m p a r e d with the m a x i m u m possible 7 x 5 = 35 strand crossing n u m b e r 476 of the (14, 35) resonant torus knot having genus 521.

6. Conclusions M u c h interest in dynamical systems theory has recently centered on renormalization group methods, the best k n o w n application being to period-doubling cascades (Collet and E c k m a n n [25], L a n f o r d [46]). However, such renormalization and scaling theories can be extended to rather general period-multiplying

40

Ph. Holmes / Knotted periodic orbits in suspensions of Smale's horseshoe

sequences (Derrida et al. [32, 47]). In this respect, the construction of the subtemplate Za(r°o)c.,~rt and the iterated horseshoe knots which lie on it is a knot theoretic analogue of the renormalization group transformation. Just as fq01+0 exhibits all the bifurcations of f~l + 'in miniature', any ql-periodic knot (or link) on X'vi has its analogue as a qoql periodic iterated horseshoe knot on *W(rq°0). For example, the kneadinginvariants + - - + - - + * + - - + - - + and + - - + - - + . + - - + - - + . + correspond to period 49 knots which are iterations of the (7, 3, - 2 ) period 7 pretzel xy2xy 3. While it seems difficult to produce general formulae analogous to those of theorem 5.2, it should be possible to compute linking numbers for at least some such non-cabled knots. These remarks suggest two directions in which the present work might lead. Picking 'regular' (e.g. t we can construct families of 'self-similar' knots, periodically repeating)defining sequences v°° * .-- * rq,, those of corollaries 4.5 and 5.4 being merely the first such examples. Such families may help one understand the total organization of the horseshoe knots. Secondly, since linking numbers and knot types are topological invariants, if one can systematically compute linking information and knot types for specific families of orbits as in sections 4 and 5, then one has considerable control over the kinds of bifurcations which can occur for parameterized families of flows which bifurcate to horseshoes. For example, if in the 'complete' horseshoe two k-periodic orbits K x and K 2 are non-isotopic knots, then they cannot annihilate one another or meet in a bifurcation. Similarly, if g 1 and K 2 are isotopic, but are linked in a different fashion about a third orbit K 3 (I(K~, K3) * I(K 2, g3)), then once again they cannot meet in a bifurcation. This ultimately leads to information on the genealogies of orbits. See Holmes and Williams [2] for examples.

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