Knotted periodic orbits in suspensions of smale's horseshoe: Extended families and bifurcation sequences

Physica D 40 (1989) 42-64 North-Holland, Amsterdam

KNOTTED PERIODIC ORBITS IN SUSPENSIONS OF SMALE'S HORSESHOE: EXTENDED FAMILIES AND BIFURCATION SEQUENCES Philip HOLMES Departments of Theoretical and Applied Mechanics and Mathematics and Center for Applied Mathematics, Cornell University, hhaca, NY 14853, USA Received 24 January 1989 Revised manuscript received 17 June 1989 Accepted 17 June 1989 Communicated by R.S. MacKay

We consider knotted periodic orbits in the "natural" suspension of Smale's horseshoe map as constructed by Holmes and Williams (1985) and Holmes (1986). We identify quartets and octets of periodic orbits which are isotopic knots and indicate how they are related via the sequences of bifurcations in which they are created in two-parameter families such as the H6non map.

1. Introduction This is the third in a series of papers in which we study a suspension of Smale's horseshoe map with a view to classifying the knot types of periodic orbits (see refs. [1, 2]). Williams [3, 4] and Birman and Williams [5, 6] observed that periodic orbits of three-dimensional flows, being dosed curves, could exhibit non-trivial knots and links. More significantly, they described a knot-holder or template Construction which enables one to systematically enumerate and build all the knotted periodic orbits contained in the hyperbolic chain recurrent set of any three-dimensional flow possessing one. They applied this idea to the geometric Lorenz attractor and showed, among other results, that it contained infinitely many non-isotopic toms knots as well as iterated toms knots and other, non-toms knots and links. They also considered several other hyperbolic sets, including the horseshoe (cf. refs. [7, 8]). Hyperbolic sets, and the horseshoe in particular [9, 10], arise naturally in periodically forced nonlinear oscillators such as the Duffmg, Van der Pol and pendulum equations [11]. It is now well known that many such ordinary differential equations apparently possess strange attractors for certain parameter values and that they certainly do possess complicated attracting sets containing horseshoes. Horseshoes in a sense provide the "backbone" for the attractors and it is thus of interest to describe the manner in which they, and the infinite families of periodic orbits within them, are created in sequences of bifurcations as the attracting set is built. Observing that knot types of periodic orbits cannot change in parameterized three-dimensional flows, Holmes and Williams [1] proposed that these topological invariants be used, with local bifurcation theorems for one-dimensional and area preserving maps, to identify orbits and establish "genealogies." Holmes and Williams [1] obtained existence and uniqueness results for pairs of resonant toms knots in the horseshoe suspension: orbits of period q which are (p, q) toms knots for 0 < p < q. Such orbits appear 0167-2789/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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

43

naturaUyin Hopf bifurcations of area preserving maps, but the template construction permits one to relate them directly to a one-dimensional unimodal map and hence to use the kneading theory (cf. ref. [12]). In this way it was possible to show that, as one passes from the one-dimensional (strongly dissipative) limit to the area preserving (Hamiltonian) case, the resonant torus bifurcation sequence is precisely reversed. Loosely speaking, this implies that there are infinitely many routes to chaos [13]. Holmes [2] then studied period multiplying sequences, of which Feigenbaum's [14] period-doubling sequence is prototypical, and showed that the knots thus created are all cables. Among them are infinitely many, non-isotopic, uniquely determined iterated torus knots. However, the cables are themselves special cases of more general iterated horseshoe or template knots, which are in turn examples of the satellite-companion construction of knot theory (ref. [15], p. 111). This work, together with related results due to Williams [7], Franks and Williams [8], Bedient [16], Holmes [17] and others, is summarized in the review by Holmes [18]. It can be seen as a contribution to a general question which has both independent interest and deep implications for orbit genealogies and bifurcation sequences: the classification of all knots riving on a given template, say that of the horseshoe. One already knows that there are infinitely many non-isotopic knots (cf. ref. [8]), but that some are excluded: for example, there are no (p, q) torus knots of any period for p > {q, and only knots which can be presented as positive braids exist (ref. [1], theorem 6.1.2). Ideally, one wants a way of passing directly from the symbolic dynamical description of an orbit as a word in two letters to a powerful knot invariant such as the Jones polynomial [19, 20]. While this has been done in certain cases (cf. ref. [21]) it is as yet a piecemeal process. The torus and iterated torus knots we have studied all have the property that they maximize or minimize the genus, a well known knot invariant. This fact is used in the uniqueness proofs, in which we get away wit h only using three simple invariants: genus, braid number and the (dynamic) period. However, it is easy to produce examples of orbits which share these invariants and yet are non-isotopic, for example by computing their Alexander or Jones polynomials (cf. ref. [18], §11). One is naturally led to ask how many distinct isotopy classes of orbits exist for each period k among the N ( k ) - 2 k / k orbits of that period. While polynomial invariants can prove that knots and links are nonisotopic, being incomplete they cannot establish isotopy and one must usually proceed piecemeal , using Reidemeister moves on specific presentations (cf. ref. [1], §8.2). In this paper we use the simple observation that a periodic orbit for the forward flow el is also periodic for the reversed flow ¢_ t to characterize the words of certain groups of orbits which are isotopic knots. In ref. [1], proposition 8.2.1 and theorem 8.2.5, it was proved that "most" horseshoe knots come in isotopic pairs, related to saddle-node bifurcations in an associated one-dimensional map, and one family of isotopic quartets was found (ref. [1], proposition 8.2.1 and theorem 8.2.2). Here we describe an infinite heirarchy of such quartets and show that, if additional criteria are met, the orbits actually come as isotopic octets. These are the extended families of our title. The forward and backward template construction, a modest extension of Birman and Williams [5, 6], is given in section 2 and the main theorem in section 3. In section 4 we use these genealogical results, with a symmetry of the Hdnon map, to augment the bifurcation reversal results of Holmes and Williams [1]. In this respect we note that our isotopic families include certain orbits proved isotopic by Sannami [22] and that our results appear to settle his conjectures on "non-satellite sequences and cusp connections" (cf. ref. [23]). We also show how some of our families are related to certain homoclinic orbits to a fixed point created by bifurcations in the "primary homoclinic fan" of Holmes and Whitley [24]. Holmes [2] used the * -product decomposition of kneading theory [25-27] to produce families of iterated horseshoe knots, including the isotopic cables and iterated torus knots mentioned above. We describe in

44

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

section 5 how additional families of isotopic, iterated quartets and octets follow from this construction. Finally, in section 6, we give a number of examples of extended families. For the basic ideas used here, including the template construction, see refs. [5, 6] or refs. [1, 2]; the last two also contain brief reviews of the horseshoe construction, symbolic dynamics and the requisite kneading theory. For more background in knot theory, see ref. [15] or ref. [28]; in dynamical systems and hyperbolic sets, ref. [11].

2. Forward and backward templates and kneading invariants 2.1. Templates Let %: M 3 ~ M 3 be a flow on a three-manifold having a hyperbolic, chain recurrent set 12 with a neighborhood N c M 3 [29, 11]. Let -+ (resp. =-) denote the equivalence relation: z 1 -+ z 2 if [Irpt(Zl)cp,(z2)ll ~ 0 as t ~ + oo and ¢pt(zi) ~ N; Vt > 0 (resp. z 1 - z 2 if II~o,(zl) - ~o,(z2)ll ~ 0 as t --* - oo and + cpt(zi) ~ N, Vt < 0). Equivalently, z 1 - z 2 (resp. z I : z2) if both z 1 and z 2 lie in the same connected component WS(z) n N (resp. WU(z) n N) of some local stable (unstable) manifold of a point z ~ 12. Note that the leaves of W~(cpt(z)) and WS(%(z)) are each two-dimensional. Under -+ the flow rp, on M 3 becomes a semiflow ~0~ on a branched two-manifold K + c M 3 and under - it becomes a semiflow opt- on a second branched two-manifold K - c M 3. The pairs ( K ÷, ~ot+), ( K - , ¢Pt--) are the forward and backward templates. Effectively, -+ collapses along strong stable manifolds and identifies orbits with the same futures, while - collapses along strong unstable manifolds and identifies orbits with the same pasts. Birman and Williams considered only the forward template, but the generalization of their result is obvious:

Proposition 2.1. The collapsing maps ~ and - are each one to one on the union of periodic orbits in 12 [3-6]. Thus the knot and link types of periodic orbits in 12 survive unchanged in the template representations. In fact, even more information about (12, opt) can be reconstructed from ( K ±, opts); hence the name " t e m p l a t e " [8] is now preferred to the earlier "knot-holder" of Birman and Williams [5, 6]. Fig. 1 shows the construction of ( K +, opt+) and ( K - , opt-) for the "natural" suspension of the horseshoe that we consider in this paper (cf. refs. [1, §2] and [2, §3]). We note the geometrically obvious, but very useful fact:

Lemma 2.2. ( K ÷, rpt+) and ( K - , ~Pt-) are homeomorphic. Proof. Pictorially, simply observe that ( K - , qot-) is obtained from ( K ÷, qo~) by rotation through ~r. More systematically, let a(z) = ( a j ( )z} j ~o~_ ~ ~ (x, y }z be the bi-infinite symbol sequence of a point z ~ 12 obtained by setting

aj(z)

=

x

=y

if FJ(z) ~ Hx if FJ(z) ~ H y ,

(2.1)

where F: S ~ R 2 is the Poincar6 or time-1 map for the flow and Hx, y are the substrips of the square S

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

t I[

~

45

K*,~, )

FI ~t

~

/

K-, ~'t-) Identify t--O,-+l to obtoin "closed" templotes.

/

Bronch

line

Branch-line

/

II

J--.%// /

Fig. 1. Forwardand backward templatesfor the horseshoe. indicated in fig. 1. Note that F(Hx, y) = Vx, y and F -1 is the time-1 map of the reversed flow i~0_t, satisfying F-X(Vx, y) = Hx, y. Let 0o

A --- I2 n S =

A FJ(Uxuny) j ~ --00

be the hyperbolic set of F. Standard methods [30, 11] show that F l a (resp. F-Xla) is topologically conjugate to the left shift o (resp. fight shift 0 -1) on the space {x, y}Z:

(a(F(z)))j=aj+l(z)

or

a(F(z))=o(a(z)).

(2.2)

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

46

\/ i ' /

Y

t

1dec

line

// t

x

y

(a)

i

x

(b)

y

(c)

I

(d) Fig. 2. (a) A knotted orbit on (K +, ~+): x2y2xy. (b) The reversed orbit, x2yxy 2, on (K-, opt-). (c) x2yxy 2 on (K +, qot+). All three closed orbits are (2, 5) torus knots of period 6, as indicated in (d), which shows the isotopy moves taking the presentation of (a) into (c).

In this context -+ identifies orbits with sequences satisfying aj = bj for all j > 0 and -~ identifies orbits with aj = bj for j < 0. Since a periodic orbit is uniquely specified by a finite segment of either the forward ( j > 0) or backward ( j < 0) subsequences, it should be clear that L and z. are each one to one on periodic orbits. In particular, the (non-invertible) maps f+, f - induced by (K ÷, ~t+), ( K - , qgt) on their branch lines are topologically conjugate to the shifts o and 0 -1 on the spaces {x, y } z÷, (x, y}Z- of forward and backward going semi-infinite sequences. Since the latter are clearly themselves conjugate, it follows that ( K +, q~t+ ) and (K-, q~7) are conjugate, as claimed. [] The Birman-Williams construction and its extension given above implies that each finite, acyclic word (or collection of words) in the letters x, y corresponds to an oriented knot (or link) on each of the templates. The arrow of time provides the orientation. The word determines the knot and vice versa. Passage from template knot to word is easy: one simply follows the orbit as t increases and reads off the letters x, y appropriate to each passage over the branch. Fig. 2 gives examples. Passage from words to knots is harder and requires alphabetization of the word and its shifts using the invariant coordinates of the kneading theory. This proceeds as follows. The inoariant coordinate of a word w = w 0 . . . Wk_ 1 is defined as O(w)= 0o... Ok_ 1, where 0j =wj if the y parity of w0... wj_ 1 is even and Oj = ~ if it is odd. Here ~ j = y (resp. x) if wj= x (resp. y). Thus, for example, O(xxyxyy)= xxyyxy. Let ~> denote the natural lexicographical order: x . . . t>y . . . , x x . . . t> xy . . . . etc. Then if z z, z 2 are (periodic) points on the branch line which lie in the chain recurrent set of ( K +, q0f ) and the words of the orbits passing through

Ph. H o l m e s / K n o t t e d p e r i o d i c orbits in suspensions o f Smale's horseshoe

47

them are w(zl), w(z2), it is easy to prove:

Proposition 2.3. z t < z z ~ O(w(zx)) ~>O(w(z2)) [12]. We explicitly exhibit the word shifts and invariant coordinates for x2y2xy (fig. 2a): Word and shifts

Invariant coordinate

Order ( = permutation)

w = xxyyxy a(w) ~ xyyxyx 0 2( w ) = y y x y x x a3(w) ~ y x y x x y o a( w) = x y x x y y o 5(w) = y x x y y x

0(w) ~ xxyxxy O( o( w)) = x y x x y y 0 ( o 2 (w)) = y x x y y y 0(o3(w)) = y y x x x y 0(ti4(w)) = x y y y x y 0(o5(w)) = yyyxyy

0 1 3 4 2 5

(o6(w) = w) In this way one determines the permutation induced on the branch line. It is then a simple matter to connect the branch points in the correct order by strands lying on the template and so produce the knot. More examples can be found in refs. [1, 2, 18]. The word is also called the itinerary of the orbit, after the terminology used in one-dimensional maps [31-33]. Although the procedure sketched above is mechanical, it is in general very hard to recognize when two orbits with words Wl, w2 are isotopic knots. However, the fact that ( K ÷, %÷) and ( K - , ¢p,-) are homeomorphic (lemma 2.2) implies that the knots corresponding to the orbits with word w = w0... w~_ 1 for ¢Pt and that with the same word w for qo_t are isotopic. Specifically, let "t~,3'2 ~ M3 be two (knotted) k-periodic orbits for ~0r Observe that the collapsing maps ~ and - yield two presentations of Yl and Y2 as conventional knot diagrams [28] (fig. 1). Let Yl project via -+ to a presentation on K ÷ with word w~ = w0... wk_ 1 and Y2 project via - to a presentation on K - with word WE= wk_l... W0. Since w2 = w~-~, the presentations are identical by lemma 2.2, and consequently the two knots y~ and Y2 are isotopic:

Proposition 2.4. If Yl and Y2 are two horseshoe knots of period k with words wl = w 0 . . . Wk_1 and w2 = Wk_l... w0 = Wl 1, then y~ and )'2 are isotopic as knotted closed curves in g 3. We develop the consequences of this simple observation in the next section. Before proceeding we note a convention. To any finite word there corresponds its periodic extension and hence a (knotted) periodic orbit. However, all the shifts of a word clearly correspond to the same orbit, We therefore work with equivalence classes. For definiteness, we call the word or itinerary of the orbit that shift which is maximal, with respect to lexicographic ordering, among all shifts. For example, x2y2xy is maximal among its shifts (see above), as is x2yxyx2y, but x2yx2yxy is not ( 8 ( x 2 y x y x 2 y ) = xxyyxxxy t> O(x2yx2yxy) = xxyyyxxy). We also require our words to be acyclic, so that the word of length k corresponds to an orbit of least period k: thus x2yx2y is illegal, since it has length 6 but period 3. 2.2. Kneading theory We must recall the construction of the kneading invariant and its use in determining bifurcation sequences in unimodal maps, the canonical example of which is the quadratic function: o

o) = .-

o 2.

(2.3)

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

48

See refs. [25, 11] for general background and refs. [24, 2] for outlines relevant to the present context. If f is a unimodal map which takes an interval ! c R into itself (f~ of (2.3) with - ¼ 3, with word w = W o . . . Wk_ 1 as the infinite sequence v (w) = 0 (xylwo...

w k_ 3 c y l w o . . . w k_ 3 c y l . . . (extended periodically)),

(2.4)

where c = y if the y parity of w o. .. Wk_ 3 is even and c = x if it is odd. The kneading invariant for the two-single-letter .words is v ( x ) = v ( y ) = x x x . . , and for the two-letter word: v ( x y ) = x y x y . . . . The relevant fact is then: 2.5. If v(w~) ~- v(w2) then the periodic orbit with word w~ is created before that with word w2 for the map f, =/x - v2, as/~ increases. Lemma

This is a special case of the more general unimodal bifurcation results described in the references cited above. Example

v ( x 2 y x y 2) = O( x y x x y x x y x x y x x y

. . . ) = xyyyxxxyyy

...

~,( x2y 3) = #(xyxxyxyxxyxyxx... ) = x y y y x x y y y x . . .

p ( x 2 y 2 x y ) = O( x y x x y y y y x x y y y y . .

. ) = xyyyxyxyyy..

.

i.e. x 2 y x y 2 appears before x2y 3, which itself appears before x 2 y 2 x y . We observe that this result implies that the pair of orbits with words w x y , wyy, for any w of length m chosen such that both are maximal and of least period m + 2, are created s i m u l t a n e o u s l y (in fact, in a saddle node bifurcation). This is always the case for orbits of odd period, but if m is even and one of the pair of words w x y , w y y is [(m + 2)/2]-periodic, then the (m + 2)-periodic orbit corresponding to the other word appears singly in a period-doubling bifurcation (cf, ref. [2], section 2.3). We will use lemma 2.5 in deriving the bifurcation sequence results of section 4, Finally, we define the *-factorization of kneading invariants, which is important in understanding period-multiplying bifurcations (of which period-doubling is the simplest). Let I,1 -- v(wl) and i,2 = v(w2) be the kneading invariants of k 1- and k2-periodic orbits having words w1 and w2, respectively. Observe that, using definition (2.4), the p~ are k;periodic, and that this simpler definition differs from the more general one used in ref. [2], cf. ref. [25]. The * -product v12 = Pl * P2 is obtained by writing k 2 copies of v1, the k~ letters in each copy being transcribed directly (x ~ x, y - , y ) at the j t h stage if the j t h letter of P2 is x and inverted (x --,y, y -o x) if it is y. The *-product is therefore klk2-periodic. The word(s) wx2 of the orbit(s) corresponding to the *-product v~2 is then obtained by reversing the procedure of eq. (2.4):

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

49

Examples: w t = x y 2, w2 = xy2xy,

vt=O(xyxyyxyy...)

= xyylxyy...

v2 = O(xyxyyyyxyyyy...

(period 3),

) --- x y y x y l x y y x y . . .

(period 5),

~'12 = vl * J'2 = x y y * x y y x y = x y y l y x x l y x x l x y y l y x x [ x y y . . . , ~'t2 = x y l x x y x y y x x y x x y x x y

Ix . . . .

and

Wx2 = x 2 y x y 2 x 2 y x 2 y x 2 y w~ = x y 2,

v I = x y y , as above,

w 2 = x y 3,

v2 = O(xyxyxyxyxy...

x2yxyZx2yxZyxy2

) = xyyxxyyx...

Jq2 = Vl * v2 = x y y * x y y x = x y y l y x x l y x x a,12 = x y l x x y x y y x x y x y y

]xyylxyy...

(period 15),

(period 4), (period 12),

lx . . . .

w12 = x 2 y x y 2x 2yx 2y alone (period 12: two doublings from period 3), w1 = xy, w 2 = x y 2, It12 =

i,~ = O ( x y y y . . . )

(period 2),

v 2 = x y y , as above (period 3),

IP1 * IP2 ~--" x y

* xyy = xy[yxlyxlxylyx

w12 = x y l x y y y x y l x wx2 = x y a x y

= xylxy...

and

....

.... x y s (period 6).

In ref. [2], section 3 it was shown that periodic orbits having words w12 whose kneading invariants v12 = v I * v2 are non-trivially factorizable have the property that, after collapse by ~ ; they are contained in a s u b t e m p l a t e (.~(vl), ~0t+) c (K +, ~t+), determined by Pl (and hence its word Wl). Moreover, the branched submanifold .oq°(vl) is a "poorly embedded" copy of K ÷ (cf. ref. [2], fig. 6 and see section 5, below). We call such orbits iterated horseshoe k n o t s . Roughly speaking, each such orbit or pair of klk2-periodic orbits closely follows a kx-periodic "core", with word wl, k 2 times before closing. In knot theoretic terms (ref. [15], p. 111), such orbits are satellites and those on which they are built, c o m p a n i o n s . Holmes (ref. [2], sections 4, 5) used the subtemplate to show that, if w2 is the word of a resonant torus knot (a (p, q) torus knot of period q), and wl is any word with an odd y-parity, the product w12 corresponds to a q cabling of w~. If w~ is itself a torus knot then the resulting satellite is an iterated torus k n o t . In the present paper we will use the properties of the subtemplate to prove isotopy results for certain families of iterated horseshoe knots.

3. P a i r s , q u a r t e t s a n d o c t e t s

This section is devoted to the proof of T h e o r e m 3.1. Let w = w0... wk, k > 1 be any word such that w x y and wyy are each maximal among their k + 3 shifts and neither is cyclic. Then the two orbits with itineraries w x y , w y y are isotopic knots. Let a = a l . . . a k, b = b l . . . b k be any two words of length k > 0 and let w = W o . . . wm -- w - 1 = w i n . . , w0 be a

50

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

palindrome such that the four words wycacy, wyca-lcy, wycbcy, wycb-Xcy where c can take the values x or y, all are maximal. Then the four orbits with itineraries wyxaxy, wyxayy, wyyaxy, wyyayy are isotopic knots• If b = a - l ( b i = a k + l _ j , j = l . . . . . k) but b ~ a ( a j ~ b j , some j ) then the eight periodic orbits with ltmeranes wy y a yy, wy y b yy are ,sotoplc knots. •

.

•

-x

x

- x

x

.

.

Remark. The simplest example of a leading palindrome is w = x m, with m > k + 3. Admissibility can be 2 XX ,, violated without losing everything. For example, consider x y yyyta = 0). x2y 4, x2y2xy and x2yxy 2 are isotopic period-6 knots (cf. fig. 2 and table 8.3 of ref. [1]), but since x2yx2y is of period 3, here we have only a triplet.

Proof. The first statement is essentially theorem 8.2.5 of ref. [1]. We recall the proof. Since wxy and wyy are both maximal w0 ---x necessarily and the strand connecting the point z 0 with word w to that with word o(w) starts at the extreme left of the template in both cases. We must place the points Zk+1 with word ak+l(w) and zk+ 2 with word ok+2(W) = a-l(W). While this can be done using invariant coordinates x k X and the fact that O(Wyy) t> 8(o (Wyy)), Vk --#O, it is easier to use the geometrical properties of the map f induced by ( K +, ¢p+) on the branch line, fig. 3a. It is clear from the figure that, if z 0 is the left most point among all the images fJ(Zo),fk+2(zo)=Zk+z is the right most and that, since ok+l(Wyy) = x yyW, fk+l(Zo) = Zk+ ~ must lie on opposite sides of the gap in the branch line for wxy and wyy and that

70

ywp. Zk+l W yy

Zk+ 2

Zo

Zk+i

Zk+ 2

w_xy

(b)

Fig. 3. (a) Iterates of z o under f , (b) pairs of isotopic knots.

Ph. Holmes/ Knottedperiodicorbits in suspensions of Smale's horseshoe

51

there can be no images fJ(Zo) between these two points for either wxy or wyy (this was explicitly assumed x in ref. [1], theorem 8.2.5), but it follows from the maximality of Wyy). Since there are no intervening strands starting between these two points, the strand connecting z k + 1 to z k + 2 on the template can be lifted from left to right and vice versa. This is a trivial isotopy move. See fig. 3b. To prove the second statement, observe that if wyxaxy is periodic for tpt then y x a - t x y w - 1 = o , " + l ( w - l y x a - l x y ) = o,.+l(wyxa-~xy) - wyxa-lxy is periodic for ¢P-r Since ( K +, q0+) is homeomorphic to ( K - , opt-), we can apply the first statement of the theorem to produce a pair of isotopic knots on ( K - , opt) with words wyxa-lxy, wyxa-lyy. But since these orbits are 1 : 1 projections, under = , of a pair of periodic orbits of the flow q0t in M 3 (run backwards) and the same orbits project under L to orbits with the reversed words, it follows from proposition 2.4 that the pair of orbits with words (wyxa-~xy)-~ wyxaxy and (wyxa-~yy)-1 wyyaxy are isotopic knots for q0t. But applying the first statement again, each of these is itself a member of an isotopic pair wyxaxy, wyxayy; wyyaxy, wyyayy. This yields our quartet. We have already observed that two periodic orbits of q0t with words wa and wb = w~-1 are isotopic knots. To prove the final statement of the theorem, we let wa= wyxaxy, w b = y x a - l x y w -1 = y x b x y w = o,.+l(wyxbxy) - wyxbxy. Thus the knots with words wyxaxy and wyxbxy are isotopic ( i f a = b, they are identical!). Application of the second statement of the theorem shows that each of these belongs to a quartet and so yields our octet. [] - -

m

X

X

Examples. x y y X X y y ; ,"

X

X

m>5

(a=xx)

are a quartet of (2, 9) torus knots of period m + 6 > 1 1 ; m

X

X

m

x

x y y y y y y ; m > 2 (a = y y ) are a quartet of(3, 7) torus knots of period m + 6 > 8 and x y y y x y y , x y y x y x y y ; m > 3 (a = y x , b = xy) are an octet of (7,3, - 2) pretzel knots of period m + 6 > 9. See section 6 below. Remark. If v = Vo... v,. =Vm... V0 then the word v is a palindrome and by extension we .call its orbit a palindrome. If v ~ v -1 then we call the two distinct orbits with words v, v -1 mirrors. Note that each quartet of theorem 3.1 contains two palindromes (wyxaxy, wyyayy) and a pair of mirrors (wyxayy, wyyaxy), while each octet contains four pairs of mirrors (wy2ay 2, wy2by2; wyxaxy, wyxbxy; wyxayy, wyybxy; wyyaxy, wyxbyy), and no palindromes. Once we have a quartet or octet of orbits we can add additional "trivial" x ' s at the beginning of the corresponding words to produce further isotopic quartets or octets, appealing to Proposition 3.2. If K 0 is an n-periodic horseshoe knot with maximal word v = v0 . .. On-1 then for each 1 < k < oo, the word xkv belongs to a (k + n)-periodic knot, K k, isotopic to K 0 (ref. [1], proposition 8.2.4). Proof. Since the strand connecting the point z0, with word v, to z 1, with word a(v) = Vl... o,._1oo lies at the extreme left of the template, k additional trivial loops can be added to accommodate the strands connecting the points with words xkv, xk-lVX . . . . . Ol... Om_lVOxk without changing the kngt types, cf. fig. 3. [] In a recent paper by Sannami ref. [22] (cf. ref. [23]) a result is obtained which is a special case of theorem 3.1. W e end this section by outlining how Sannami's result is related to ours. In terms of the

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

52

present paper, he shows that the quartet of orbits with (maximal) words of the forms X

XlClYC2y...C,jyy

and

I

X

l>_l

XdlydEy...dmyyy,

(3.1)

are isotopic if the (reversed, inverted) sequences ?,~... cl, Jm.-. dl are 2ith, (2i + 1)st neighbors with respect to ordering of invariant coordinates of the 2 m words x m through ym (ref. [22], theorem 3). In this respect, we have the following: Proposition 3.3. Any two such neighbors have orbit words of the forms xty2(m-k-2)xyXvfkY.., flyXvy for some 0 < k < m - 2, and word f = f t --. fk. Hence, for l = 1, they form a quartet with w = xy2(m-k-2)X, a = Y f k Y ' " fxY and are isotopic by theorem 3.1 directly. If l > 1, they are isotopic by theorem 3.1 and .I

J

proposition 3.2 applied with k = l - 1. Proof. The first and last words of length m with respect to lexicographical order are x " , ym, so that any neighboring intermediate pair, say the 2ith and ( 2 i + l ) s t , have the forms e l . . . e k x Y m-k-1 and e l . . . e k y y m - k - ~ for some 0 < k __
d 1 ... d,n into the full words (3.1), we obtain the words of the proposition.

[]

Example. Let m = 3 and order x x x t> x x y t> x y x e, xyy ~, y x x ~, y x y t> y y x v, yyy, being the invariant coordinates of the words x x x , (xxy, xyy), (xyx, yyx), (yyy, y x y ) and y x x respectively, where we have

bracketed the pairs (2, 3), (4, 5), (6, 7). Reversing and inverting, this gives three quartets: (2, 3)

x x y , xyy =, xyy, x x y =' x Ix y yXy y y yXy :

(4, 5)

x y x , y y x = ' y x y , yXX

(6, 7)

yyy, y x y =, x x x , x y x ~ x lxy yX y x y yX y:

w=

XI+I '

a = y3;

X ty y X y Xy y yXy : l v = x l y 2 x ' a =Y; tg = X I+1,

a = yxy.

See ref. [22], fig. 1.5.6. Sannami calls words of the form (3.1) satellite sequences, since the corresponding orbits in the unimodal map v ---,/t - v2 lie "near a hyperbolic set of period 2". Such orbits are related to kneading invariants which can be factored under the *-decomposition as described in ref. [2]; also see section 5 below. He also raises the question of the relation between satellite and non-satellite words and the corresponding isotopic quartets and "cusp connections" in the bifurcation set of the orientation preserving Hrnon map (ref. [23], section 3.1). Since our results include many non-satellite sequences, we are able to (partially) settle his conjecture. We discuss these bifurcation results in section 4.

4. Bifurcation sequence reversals in the H~non map

In this section we extend the bifurcation sequence results of Holmes and Whitley [24] and Holmes and Williams [1], using the preceding observations on palindromes and mirrors in the specific case of the H r n o n map. In the first part of the section we give rigorous results on bifurcation sequences in certain

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

53

limits. We close with looser speculations on the implications of these results for the global structure of the bifurcation set. We take the H~non map in the form

(u. v)

F .g u. v)= ( o . -

+ , - o7.).

(4.1)

although the quadratic function / ~ - o 7. can be replaced by any even, unimodal function f , ( v ) with negative Schwarzian derivative [34] without affecting our results. Note that the usual Hrnon map (x, y ) ~ (1 + y - ax 2, bx) can be transformed to (4.1) for any a ~ 0. We concentrate on the orientation preserving case, e > 0 (b < 0), since we are interested in suspensions of the horseshoe which are flows on orientable three manifolds, but some of the results below apply to E < 0 also. If e q: 0, (4.1) can be inverted to give -1 F;..(u. o) = ( ( - o +

(4.2)

and it is easy to verify the following:

Lemma 4.1. If {(a,,b,)},~=_~ is an orbit of F~,,, for e4.0, then {(b,,/e,a,,/e))~.=_~ is an orbit of F~/le2,1/e" In the special case of the area preserving map det DF~, ~ = e -- 1; this implies that backward orbits of F~,1 are reflections of forward orbits of F~,1 in the line u = v. Holmes and Whitley (ref. [24], proposition 4.3) used this fact to prove that there is an infinite set of "cubic" homoclinic tangeneies as # increases for e = 1. We return to these below. In the limit e ---, 0 the behavior of (4.1) is increasingly dominated by that of the one-dimensional map v ~ # - v 2. In particular, a simple application of the implicit function theorem yields

Lemma 4.2. For any N < oo, there exists ~(N) > 0 such that the sequence of bifurcations of periodic orbits of periods k < N for F~,~ as/~ increases for fixed e with [el < e(N), is identical to that of the map v ~ # - o 2, and so is determined by kneading theory [35]. This result, together with lemmas 2.5 and 4.1, implies that we can also determine bifurcation sequences of F~,, in the "strongly expanding" limit e -~ oo. To make for a manageable description of bifurcations, we must agree on a convention for identifying orbits in the "incomplete", non-hyperbolic horseshoe. Devaney and Niteki [36] proved that, for # > 1(5 + 2~/3-)(1 + le12), F~,~ has a hyperbolic horseshoe and consequently that the periodic orbits are uniquely determined by their words, as in the usual symbolic dynamics, or, indeed, as in the template construction of section 2. As # decreases and periodic orbits begin to vanish, it is no longer possible to do this, since a Markov partition cannot generally be defined. Nonetheless, if we follow a branch of periodic orbits as/x decreases from the hyperbolic region, say for fixed e, we can continue to identify the orbits in (the connected component of) the branch by the word that describes their itinerary in the hyperbolic region. This can be done uniquely provided no other branch crosses the chosen one. (Branches that terminate on the given branch, in period-multiplying bifurcations for example, do not affect its identity.) In the situation

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

54

II v II

wxy

~xy

wyy

--

wyy

Hyperbolic 2 s h i f t

Fig. 4. Identification of periodic orbit branches.

of lemma 4.2, branches cannot cross, since one is close to the one-dimensional case, and we are consequently justified in identifying orbits created in saddle-node bifurcations by their "eventual words". See fig. 4. If branches do cross we must establish a convention to deal with this. Supposing that we have done so, we can now state

Lemma 4.3. Suppose that (/~, e ) = (a,/3), 0 < fl _< 1 is a saddle-node or pitchfork bifurcation point at which a periodic orbit with (eventual) word w is created for F~,~. Then (ix, e)=(a/flE, l/fl) is a saddle-node or pitchfork bifurcation point at which a periodic orbit with word w-1 is created for F~, ~.

Proof. Let the orbit y+ created at (a,/3) be given by {(a,, b,)),_ k-1o where k is the period. Then, by lemma 4.1, F~-~ has an orbit ((bJfl, a,u/fl)}~- ~ at (~t, e ) = (a/fl2,1/fl) and so F~, e has an orbit 3,- = ((bk_./fl ' ak_,/fl)}..k 1. In the hyperbolic region the word w + of 7 + is determined by the sequence { b , } ,k_- 1o of v coordinates, since ( a . , b.) lies in Hx if b, < 0 and in Hy if b, > 0. Similarly, the word w- of 3' - is determined by the sequence ( b k _ n / f l } nk-1 . 0 of u coordinates, for (bk_n//fl, ak_n/[~) lies in Vx if bk_n//fl < 0 and Vy if b k _ n / f l > 0. S i n c e fl > 0, t h i s implies that w - = (w+) -1, as claimed. T o verify that (a/fl2,1/fl) is a saddle-node (resp. pitchfork) bifurcation we simply observe that, if (/~, e) = (a, fl) is a saddle-node (resp. pitchfork) for F~,, then lemma 4.1 implies that (/~, e) = (a/fl 2,1/fl) is a saddle-node (resp. pitchfork) for F~.~, and consequently also for F~, ~. [2 Corollary 4.4. If an orbit w is created in a saddle-node or pitchfork bifurcation at (~, e) = (a, 1) then its mirror partner w-1 is created simultaneously. We can now state the main result of this section.

Theorem 4.5. (1) Let w --- w-t, a = a-1 be palindromes of length m + 1 > 1 and k > 0 respectively, chosen so that the X X four words wYvavY are all maximal and of least period n = m + k + 5. Then there exists e(n) > 0 such that, as ~ increases for fixed e x (a) If the y parity of w is even and 0 < e < e(n) the quartet is created in the sequence wyxayy, J

J

X X X ~:,yayy and, if 1 / e ( n ) < e < ~ , in the sequence ~yyaxy, wyyayy.

(b) If the y parity of w is odd and 0 < e _< e(n) the quartet is created in the sequence and, if 1 / e ( n ) N e < ~ in the sequence

X

X

wyyayy, ~/yaxy.

X

X

wyyayy, wyxayy

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

55

(2) Let w = w-x be a palindrome of length m + 1 >_ 1 and a ~ b, a = b-1 be a mirror pair of length k > 2 X X X.X whose invariant coordinates satisfy 0(a) ~ 0(b) and chosen so that the eight words wy y a y y, wy y a y y are all maximal and of at least period n = m + k + 5. Then there exists e(n)> 0 such that, as /x increases for fixed e, (a) If the y

parity of

X

X

w is even and O
the octet is created in the sequence

X

X

X

wyxayy, wyxbyy, wyybyy wyyayy and if l / e ( n ) _< e < oo, in the sequence wyybxy, wyyaxy, x

x

wy yayy, wy ybyy. x

x

(b) If the y parity of w is odd and 0 < e < e(n), the octet is created in the sequence wyyayy, wyybyy, X

X

X

X

X

X

wyxbyy, wyxayy and, if 1/e(n) < e < oo, in the sequence wyybyy, wyyayy, wyyaxy, Wyybxy. (3) As/~ increases for e = 1 the mirror pair wyxayy, wyyaxy is created simultaneously in case 1 (a = a - l ) , while in case 2 ( a S b = a -1) the four m i r r o r pairs (wyxayy, wyybxy), (wyxaxy, wyxbxy ),( wyyaxy, wyxbyy ) and ( wyyayy, wyybyy) are created in the pairs indicated. Proof. Ordering the kneading invariants and use of lemma 2.5 gives the bifurcation sequences for the one-dimensional limit v ~ / ~ - v 2 and hence,via lemma 4.2, for the map F~,~ with 0 < e < e(n) sufficiently small. In all cases the kneading invariant starts with the same m + 3 symbols O(xyw). In the case that w has even y parity, xyw has odd parity and so the (m + 4)th and (m + 5)th symbols of the kneading invariant, derived from the letter pairs "yx" and "yy" are "xx" and "xy" respectively. If w has odd parity, we obtain "yy" and "yx ". This gives the ordering for cases l a and l b with 0 < e < e(n). Fix such x an e and suppose that the two bifurcation points are/~1, at which the pair of orbits with words wyxayy x

appear, and #2, at which wyyayy appear. By lemma 4.3,

( / L 1 / e 2, l / e )

and (/~2/e 2, l / e ) are bifurcation

x -1 - w - l y y a - l x y = w y yxa x y points for pairs of orbits with words (wyxayy)

and (wyyayy) x -1 -

w_ly y a _ l y y = wyyayy x respectively. If/~l < ~2, ~1/e2 < ~2ff e2 and if/~1 > ~2, /'£1/e2 > /~2/ e2, completing the proof of cases l a and lb. Case 2 proceeds in the same way, the key observations being that, with even y parity in w:

O(xywyxa) = xy. . . xxO(a) ~, O(xywyxb) = x y . . . xxO(b) t~ O(xywyyb) = x y . . . yxO(b) t~ O(xywyya) = x y . . . yxt](a), where ~(e) denotes inversion of each entry in O(e)(x ~ y and y ~ x). Again the first m + 3 entries of each sequence are identical and the ordering determined by a and b follows from the fact that, if 0 ( a ) t~ 0(b) then ~(b) t~ ~(a). With odd parity in w we have O(xywyya) = xy... yxO(a) ~ O(xywyyb) = x y . . . yxO(b) t~ O(xywyxb) = xy... yyl~(b) ~, O(xywyxa) = xy... yy~(a). Application of lemma 4.3 to the four bifurcation points then completes the proof of cases 2a and 2b. Finally, to obtain the third statement we merely apply corollary 4.4 in case 1 to the unique mirror pair X X X X wyxayy, w lyya-lxy = wyyaxy and in case 2 to the eight words Wyyayy, w y y b y y , using the fact that w = w- 1 and b = a - 1 to select the appropriate mirror pairs. [] The bifurcation set on which a specific k-period orbit of the H~non map appears is given by the conditions that Ff~(h, ~ ) = (~, ~) and that the eigenvalue of DF~k,(~, ~ ) = +1. Since det DF~,~ = e k, the latter condition is equivalent to trace DF~k~(~, ~) = +(1 + ek). Elimination of (h, ~) from these equations yields relations between the parameters (#, e) determining saddle-node and pitchfork bifurcations in the + case and period-doubling bifurcations in the - case [11]. Since the equations are (high degree) polynomials, the bifurcation curves which the implicit function theorem and lernma 4.2 allow us to propagate from

56

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

bifurcation points of the one-dimensional map y ~ l ~ - y 2, can be analytically continued in the (/~, e) plane. Holmes and Whitley (ref. [24], proposition 3,1) use this fact to prove that, for "regular families" F,, every bifurcation curve C originating at a saddle-node point (/~, 0) on e = 0 can be continued to e = 1 so that, except possibly at isolated points where there are cusp (pitchfork) bifurcations, F,,, has a saddle-node as one crosses C transversely. We remark that it is not known that the H6non map (4.1) is regular in the sense required by Holmes and Whitley [24] and hence more degenerate (codimension > 3) bifurcation points may lie on C. It is, however, possible to prove that infinitely many pitchfork bifurcations occur as/~ increases on e = 1. x x Let w = w -~, a = a - 1 and consider the quartet of orbits w y y a y y of theorem 3.1, which contains the two palindromes wyxaxy, wyyayy. If wyxaxy (say) is an N --- m + k + 5 periodic orbit ((a,, b,) }nN-l= o under F~,1 then by lemma 4.1, the orbit {(bn, an)},_ N-X 0 is alSO N-periodic under E~-1 , 1 " Since both forward and backward orbits have the same word wyxaxy and the collapsing maps --+ and - of section 2 are each 1 : 1 on periodic orbits, we conclude that the coordinates an, bn must satisfy bn_J = a j, j = 0 , . . . , N - 1 and hence that the orbits of palindromes are symmetric under reflection about u = v. Local bifurcation theory with symmetry (cf. refs. [37, 38]) implies that, if such an orbit undergoes a bifurcation with eigenvalue passing through 1, it must be a pitchfork in which (at least) one pair of periodic orbits "ix ~ 3'2 appear which are reflections of each other over u = o. These are, of course, the mirror pair wyxayy, wyyaxy of the quartet. Breaking the reflection symmetry by perturbing e from 1 leads, under generic hypotheses, to the usual cusp bifurcation which is most easily described pictorially (ref. [11], section 7.1.). Fig. 5 shows the case in which the mirror pair bifurcates from the palindrome wyyayy; a similar picture obtains for bifurcation from wyxaxy. Theorem 4.5 immediately implies that certain pairs of (saddle-node) bifurcation curves not only extend to arbitrarily high e from e = 0, but that they cross. To get a better idea of the structure of some of these bifurcation sets and to relate them to the cusps found above, we recall results obtained by Holmes and Whitley [24], who considered bifurcations in which orbits homoclinic to the saddle point with word x are created from the H6non map. In particular, they proved Proposition 4.6. Fix k < oo and consider the homoclinic orbits of F~,~ to the fixed point x having X oo itineraries X ~ a k . . . a l y y X , where a = a l . . . a k is any word of length k. Then there exists c(k) > 0 such that, as/~ increases for fixed e with 0 < e < e(k), these orbits are created pairwise in 2 k+l bifurcations in the sequence determined by the inverse coordinates O - ( a ) = O ( x a _ l . . . a_k); i.e., if 8-(a)t> 8 - ( b ) . X oo then the pair x ooa yX X oo is created before x o ~OyyX , (ref. [24], theorem 4.8). Using results of Gavrilov and Silnikov [39, 40], they also implicitly showed the following. Lemma 4.7. Let (#, ~) be a bifurcation point at which a quadratic homoclinic tangency occurs in which a pair of orbits with itineraries x~Way Wbx ~ is created. Then (/~, e) is the limit as l ~ o¢ of a sequence of x saddle-node of bifurcations at which pairs of periodic orbits with itineraries x I Way wb appear. The observations above regarding pitchfork bifurcations of periodic orbits of the form x+yyayy, xlyxaxy obviously extend to homoclinic palindromes x~yyayyx °°, x~yxaxyx~: in fact, these are among the "cubic" homoclinic bifurcations of Holmes and Whitley (ref. [24], proposition 4.3). Moreover, if, as numerical computations suggest, the quartet of homoclinic orbits x y yXa yXy x oo are created in pairs

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

57

_wyx_axy - p a l i n d r o m e s

_wyyaxy- m i r r o r E=I w_yy_ayy - p a l i n d r o m e s

_wyxayy-

mirror

(b)

(a)

yya~ y

(c)

~<1

_

_ayy

~>1

Fig. 5. C r e a t i o n of a m i r r o r p a i r for a p a l i n d r o m e in a cusp bifurcation. (a) Bifurcation set; (b) b i f u r c a t i o n d i a g r a m o n (c) generic p e r t u r b a t i o n s for r < 1 and r > 1, saddle-node pairs indicated.

~

= 1;

X oo x ooy x a yX y x oo, x°°yyayyx as # increases for all fixed e < 1, and these two bifurcation curves do not cross, then lemma 4.3 and lemma 4.7 together imply that they are created pairwise in the sequence x ooy yXa x y x oo, x ocy yXa y y x °~ as # increases for fixed e > 1. The cusp observations and the ordering results of theorem 4.5, for periodic orbits, and proposition 4.6 and lemma 4.7, for homoclinic orbits, then enable us to draw the conclusions shown in fig. 6. Here the bifurcation curves are identified by the central blocks xax, xay, yax, yay of the words of the orbits created as they are crossed transversely. In interpreting this figure we note that there are two independent asymptotic results at work. The kneading invariants of l X I X x y x a y y , x y y a y y determine the order of bifurcations of periodic orbits of fixed period l + k + 5 as e ~ 0 ÷ (and ~ o¢) (theorem 4.5), while the inverse coordinates #(yxa), O(yya) determine the order of bifurcation of periodic orbits for fixed e > 0 as l ~ oo (proposition 4.6 and lemma 4.7). This accounts for the fact that the saddle-node bifurcation curves can cross in the case of odd y parity in a (fig. 6b). We remark that, had we assumed that these saddle-node bifurcation curves did not cross in either case, then it would follow that the homoclinic bifurcation curves would cross in the case of odd y parity. These observations provide a partial answer to Sannami's [22, 23] conjectures on isotopic quartets and cusp bifurcations. We now move on to consider the bifurcation sets of the isotopic octets of theorem 3.1. In the case of octets, no palindromes exist and we cannot directly infer the existence of pitchfork bifurcations and the cusp structure of figs. 5 and 6. However, if we assume that the four saddle-node X X X X bifurcation curves of theorem (4.5) on which orbits with words wyxayy, wyxbyy, wyybyy, wyyayy are

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

58

O_Xyj/

E

/

¢/

/,i.s

0

I

o

F i g . 6. P e r i o d i c ( ) a n d h o m o c l i n i c ( - - - ) b i f u r c a t i o n sets f o r q u a r t e t s o f i s o t o p i c orbits. (a) a h a s e v e n y p a r i t y , (b) a h a s o d d y p a r i t y , w = x t h a s even y p a r i t y in b o t h cases.

created for small e continue without passing through cusps or other codimension >_ 2 bifurcation points to e = 1 we arrive at a remarkable conclusion. For by lemma 4.3 the analogous saddle-node curves for orbits x X x X with words wyybxy, wyyaxy, wyyayy, wyybyy likewise continue from e >_ 1/e(n) to e -- 1. Corollary 4.4 then implies that the four saddle-node curves must meet at a single point (/~*,1), on e = 1, at which the four mirror pairs with central blocks ( xax, xbx ), xay, ybx ), ( yax, xby ) and ( yay, yby ) are simultaneously created. The existence of such a point (of codimension >_ 4) is highly non-generic in a two-parameter family such as F~,,r Assumption of a single cusp point on one of the four bifurcation curves for e < 1 (and thus another for e > 1, by lemma 4.3) makes for a more reasonable conjectured structure, with only codimension-2 bifurcations, cusps and pairs of saddle-nodes at points where bifurcation curves cross, as expected in a regular two-parameter family [24]. T o provide an illustrative example that actually does occur in the H6non family, we again appeal to the results on homoclinic bifurcations in ref. [24] (section 4.4). In that paper it was shown that a countable set of "irregular" homoclinic bifurcations occurs, successively disrupting the ordering of proposition 4.6 as e . . . . . OQ X oG increases. In particular, the four bifurcation curves on which orbits with mnerarles x yyyxyyx , x - y x y x yXy x - ,~ x -~y y x y yXy x ~ , x ooy x x y yX y x - are created have the structure indicated in fig. 7a (cf. ref. [24], fig. 19]). As in fig. 6, the orbits are identified by the central blocks of (four) symbols which distinguish them. Finally, using the small-e ordering results of theorem 4.5 and the simultaneous saddle-node bifurcations on e = 1 resulting from corollary 4.4, we can sketch the bifurcation set for the isotopic octet x iy yXx y yXy , x iy yx y x yXy as in fig. 7b. While we cannot determine the complete relative order in which the saddle-node bifurcation curves cross one another in the (bt, e) plane, our results do tell us the order of saddle nodes as e --* 0 and --* oo for fixed l and as l ~ oo for fixed e ¢ 0, oo. One could construct similar, bifurcation sets for other octets.

59

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

X

X

X

I

xxy~,

0

.x;-/ a x,x;, X

X

,-

~xyx

;yxy /

E,

~/xy ~X,, fXX/

xyy

(yxyx, xyxy)

(yxyy, yyxy) (xxyx, xyxx) (xxyy, yyxx)

xxy; yxy~, x yx.~.~yyx~/

o

f

f

f

J

xyxy~ \ X

•

yyx~

Fig. 7. Irregular homoclinic bifurcations and octets of isotopic orbits. (a) A cusp for e < 1. (b) Homoclinic (- - -) and periodic ( ) bifurcation curves, indicating simultaneous saddle-nodes creating mirror pairs on • = 1. w== x t, a = xy, b = y x . Orbits identified by the central block of four symbols.

5. Extended families of iterated horseshoe knots T h e * - p r o d u c t operation on kneading invariants, outlined in section 2, permits us to construct infinite families of iterated horseshoe knots. It is natural to ask if such families preserve the isotopies of the " c o m p o n e n t " knots. In this respect, we have the following: T h e o r e m 5.1. If K 1 and K 2 are isotopic/-periodic orbits belonging to a quartet or octet with itineraries w1 a n d w2 and kneading invariants, J'l and P2 and K 0 is a n y other k-periodic orbit with word w0 and

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

60

(a) Case A

(b) Case B

(c)

Fig. 8. The braids Bj and the connecting piece ~ of -o~a(Vo): (a) case A, (b) case B, (c) ~ is an oriented ribbon.

(1)

(2)

(3)

Fig. 9. Reidemeister moves.

kneading invariant I,0 then the knots Kot , K02 having words Wot,1¢02 whose kneading invariants ~re ~'01 = 1'0 * vl, v02 = Vo* v2 are isotopic k/-periodic knots.

Proof. The subtemplate for *-product knots, constructed in ref. [2], sections 2, 3, shows that there are two cases to consider: (A) when the number nkO of y'S in the first k entries of the word XyWo... Wk_3... from which the kneading invariant Po is computed is odd, and (B) when nko is even (cf. eq. (2.4)). The subtemplate .~a0,0) is a copy of the "splitting piece" of the branched manifold K +, closed not by a trivial ribbon as in fig. 1, but in case A by an orientation reversing (knotted) ribbon and in case B by an orientation preserving (knotted) ribbon (figs. 10 and 11, below, give examples). Since the iterated knots Kol, Ko2 he on the same subtemplate ZP(v0), the 0nly place they differ is in their passage over the splitting piece. In case B we extract the piece alone and in case A we extract the splitting piece plus a half twist, calling the resulting (open) /-brands B 1 and B 2. Note that the remainder of L~a(v0), denoted ~t, which carries the l parallel strands connecting the bottom of Bj to its top, is now an oriented (knotted) ribbon in both cases. We finally form closed braids Bj from Bj in the trivial way: fig. 8. We claim that the resulting "induced" knots B1, B 2 are different presentations of a single knot and that B1 can be isotoped to BE by a sequence of Reidemeister moves of types 2 and 3: fig. 9 (ref. [28], p. 11). It then follows that K01 can be isotoped to K02, since the moves which suffice for B~ and B2 can be propagated around the oriented ribbon ~t without affecting the "intrinsic" effect on the strands of Kox, Ko2 (since M is oriented, its normal bundle allows one to define " t o p " and "bottom" uniquely). It remains to show that B1 and B2 enjoy the property claimed. In case B this is easy. Examination of the kneading invariant z,0j= vo * vj shows that the letters 2k - 2 +jl, 0 < j < l - 1 rood kl, of the word w0j are precisely those of wj (with the usual ambiguity of the penultimate letter) and so that Bj is an exact copy of Kj. Now K 1 and K 2 belong to a quartet or octet and so are either trivially identical (wxy, wyy; fig. 3b) or different projections (under + , - ) of the same knotted dosed orbit in M 3 (e.g. wyxayy, wyya-lxy). Since the original periodic orbit is itself a (positive) braid on l strands, the differing projections under the collapsing maps Z and :- merely interchange strand order and crossing sequences and can therefore be isotoped to each other using Reidemeister moves 2 and 3. Move 1 is not necessary, since no loops are removed or inserted in this process. In case A the braids Bj are not the same as K j, but rather correspond to those obtained by suspending the horseshoe map with an extra half twist (compare fig. 8a with fig. 1). However, precisely the same arguments as those used in sections 2 and 3 above apply to any orientation preserving suspension of the

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

s

61

/

Bo2~ Bo, Fig. 10. An example of isotopic iterated horseshoe knots. Kol and

g02

have words Wo] = x2yxy2xy2x2yx2yx~y

and %2 =

x2yxy2xy2x2yxy2x2y respectively. Isotopy on splitting piece and half twist indicated.

horseshoe, so that Bt example.

and

Example. Let w0 = xy 2,

B2 Can again be isotoped using Reidemeister moves 2 and 3. See fig. 10 for an []

x2yxy 2, SO that Po = xY 2, Pl = xy3xy, P2 = xY 3x2. Then P01 = VO * Pt = x y 2 [ y x 2 [ y x 2 1 y x 2 1 x y 2 I y x 2 ~ W01 = x 2 y x y 2 x y E x 2 y x 2 y x ~ y and Po2 = Po* P2 = xy2Jyx21yx21yx21xy21xY2 ~ %2 = x2yxy2xy2x2yxy2x2y (alone). Alphabetizing these words and their shifts we obtain the permutations 0 6 12 5 11 13 4 10 15 2 8 16 1 7 14 3 9 17 and 0 6 12 5 11 14 3 9 16 1 7 13 4 10 15 2 8 17 respectively, so that the two iterated knots lie on the subtemplate Z:'(xy 2) as shown in fig. 10. Note that the words w1 and w2 correspond to the isotopic knots /(1, K 2 shown in figs. 2a and 2c and compare the isotopy moves for the iterated knots K01, K02 in fig. 10 with that shown in fig. 2d for K 1, K 2. 1¢1 = x 2 y 4, 1¢2 ---

Remark. One cannot alter the order of *-product factors in theorem 5.1: the knots having kneading invariants rl * Po, P2 * Po are not generally isotopic. For example, let w1 = x2y 4, w2 = x2yxy 2 and w0 = xy 2, so that p x = x y 3 x y , p 2 = x y 3 x 2 and ~ o = x y 2. Then p l * p o = x y 3 x y l y x a y x l y x 3 y x l x y 3 . . . ~Wlo = x 2 y 2 x y x 2 y 4 x 2 y 2 y y and p2* I'o=xY 3x2 lyx 3y2 lyx 3y2 Ixy 3 ... ~ w20=x2yxy2x2yx2yx2yx~y. Alphabetizing

62

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

I

//:l'~~._Spl _~}it ing piece

(a) W~o

~"~Splitting

piece

(b) W2o

Fig. 11. Klo and K2o are not generally isotopic. Here w1 = x 2 y 4, w2= x2yxy 2, wo=xy 2, Wlo=X2y2xyx2y4x2y2yy, w20= 2

2 2

~

~

X

X yxy X y x - y x - y X y y . N o t e c r o s s i n g n u m b e r s Clo = 91, C2o = 89, so t h a t g e n e r a a r e 37 a n d 36 r e s p e c t i v e l y .

and computing permutations and crossing numbers, we find that the two-period 18 knots Kt0 and K20 with words wl0 and w20 each have braid number 6 but their genera are 37 and 36 respectively. These knots fail to be isotopic because the subtemplates LP(g~) and ~ ( v 2 ) , on which they lie, although knotted in the same way, have different numbers of half twists. One (wx) corresponds to case A and the other (w2) to case B in the proof above. See fig. 11. Theorem 5.1 is useful in the general problem of classification of horseshoe knots, since it implies that one need only deal with words whose kneading invariants are "prime," since those that factor as a non-trivial *-product can be understood in terms of the factors. In this respect we observe that quartets (and octets) of knots closely related to Sannami's [22, 23] satellite sequences can be produced by taking wo = x y and w~, w2 from a quartet (or octet) of theorem 3.1.

6. Summary: Some horseshoe quartets and octets Here we present a table giving the knot types or, in some cases, braid presentations, of all horseshoe 1 x x knots of the forms x y y a y y where a = ax... a k has length 0 < k < 4. It is convenient to organize these via the inverse coordinates of the homoclinic bifurcation sequences, proposition 4.6, so that they are listed in their order of creation for e > 0 fixed, sufficiently small and l sufficiently large. We note that for the "critical" word x t y x x 4 x y , and hence all other words of the form considered, to be maximal it is sufficient to take l > 7. In table 1 we list only the word a, so that each entry refers to a quartet of isotopic knots for each (sufficiently large) choice of I. In the case of a = a - 1 the quartet stands alone, if a ~ a - 1 then two quartets are paired to form an octet, as indicated. For the orbits identified by braid words, o j indicates a (positive) crossing of the j t h and ( j + 1)st strands and A k denotes a half twist on k strands. The knot and braid types were computed using the "well-disposed braid" template of Holmes and Williams [1] and a Macintosh BASIC program was used to alphabetize the words and their shifts and hence compute braid permutations and crossing numbers. We observe that there are additional isotopies not identified by l X X theorem 3.1: for example, the members of the two octets x y y a y y for a = y y y x , x y y y and y x y x , x y x y are all isotopic. We do not know yet if this and other additional isotopies are coincidental or part of a yet

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

63

Table i

a

Knot type

Braid number

xxxx xxx yxxx xx yyxx yxx xyxx

(2,13) torus (2,11) torus (11,3, - 2 ) pretzel (2, 9) toms o2(olo2) 2 As (9, 3, - 2) pretzel

2 2 3 2 3 3

6 5 7 4 8 6

o2(olo2)2a~

3

8

x

(2, 7) torus (3,10) torus (3, 8) toms 0?020302A34 (7, 3, - 2) pretzel 0~020302za34 (3, 8) torus O2(0"10"2)2 A5 (2, 5) toms

2 3 3 4 3 3 3 2

3 9 7 10 5 10 7 8 2

o2(oao2) 2 A53

3

8

(3, 8) torus 01(02o3) 3 A3 (3, 7) toms (4, 9) torus (4, 7) toms 0"20"2O3(12A~ (3, 5) torus o~o2o3o2A34 ((13, 2), (3, 2)) iterated torus O'1(ff203)3A3 (7, 3, - 2) pretzel

3 4 3 4 4 4 3 4 4

7 11 8 12 9 10 4 10 8

4 3

11 5

a3 A~4

4

9

(9, 3, - 2) pretzel (11, 3, - 2) pretzel

3 3

6 7

xyyx yyx yyyx yx yxyx xyx xxyx

O xxyy xyy yxyy yy yyyy yyy xyyy

y xyxy yxy yyxy xy yxxy xxy xxxy

4

Genus

Octets indicated

larger pattern, but the results described in this paper do go some way toward describing the global organizations of horseshoe knots. Acknowledgements This work was partially supported by NSF grants DMS 87-05581 and DMS 87-03656. I thank the California Institute of Technology and the Sherman Fairchild Foundation for providing the opportunity to complete it. References [1] P.J. Holmes and R.F. Williams, Knotted periodic orbits in suspensions of Smale's horseshoe: torus knots and bifurcation sequences, Arch. Rat. Mech. Anal. 90 (1985) 115-194. [2] P.J. Holmes, Knotted periodic orbits in suspensions of Smale's horseshoe: period multiplying and cabled knots, Physica D 21 (1986) 7-41.

64

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

[3] R.F. Williams, The structure of Lorenz attractors, in: Turbulence Seminar, Berkeley 1976/77, A. Chorin, J.E. Marsden and S. Smale, eds., Springer Lecture Notes in Mathematics, Vol. 615 (Springer, Berlin, 1977), pp. 94-116. [4] R.F. Williams, The structure of Lorenz attractors, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979) 73-99. [5] J. Birman and R.F. Williams, Knotted periodic orbits in dynamical systems I: Lorenz's equations, Topology 22 (1983) 47-82. [6] J. Birman and R.F. Williams, Knotted periodic orbits in dynamical systems II: Knot holders for fibred knots, Cont. Math. 20 (1983) 1-60. [7] R.F. Williams, Lorenz knots are prime, Ergod. Th. Dyn. Sys. 4 (1983) 147-163. [8] J. Franks and R.F. Williams, Entropy and knots, Trans. Amer. Math. Soc. 291 (1985) 241-253. [9] S. Smale, Diffeomorphisms with many periodic points, in: Differential and Combinatorial Topology, S.S. Cairns, ed. (Princeton Univ. Press, Princeton, NJ, 1963), pp. 63-80. [10] S, Smale, Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967) 747-817. [11] J. Guckenheimer and P. Holmes, Nontinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol. 42, 2nd Ed. (corrected) (Springer, Berlin, 1986). [12] J. Milnor and R. Thurston, On Iterated Maps of the Interval I and II, Princeton University, Princeton, N J, unpublished notes. [13] P.J. Holmes, Bifurcation sequences in horseshoe maps: infinitely many routes to chaos, Phys. Lett. A 104 (1984) 299-302. [14] M.J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19 (1978) 25-52. [15] D. Rolfsen, Knots and Links (Publish or Perish, Berkeley, CA, 1976). [16] R.E. Bedient, Classifying 3-trip Lorenz knots, Topology Appl. 20 (1985) 89-96. [17] P.J. Holmes, Knotted periodic orbits in suspensions of annulus maps, Proc. R. Soc. London A 411 (1987) 351-378. [18] P.J. Holmes, Knots and orbit genealogies in nonlinear oscillators, in: New Directions in Dynamical Systems, T. Bedford and J. Swift, eds., London Math. Soc. Lecture Notes, Vol. 127 (Cambridge Univ. Press, Cambridge, 1988), pp. 150-191. [19] V.F.R. Jones, A polynomial invariant for knots via yon Neumarm algebras, Bull. Am. Math. Soc. 12 (1985) 103-111. [20] V.F.R. Jones, A new knot polynomial and von Neumaun algebras, Not. Am. Math. Soc. 33 (1986) 214-225, [21] P.J. Holmes, Jones polynomials for horseshoe braids, in preparation. [22] A. Sannami, A topological classification of the periodic orbits of the Htnon family, Japan J. Appl. Math. (1988), in press. [23] A. Sannami, On the structure of the parameter space of the H~non family, in: Dynamical Systems and Applications (World Scientific, Singapore, 1987). [24] P.J. Holmes and D.C. Whitley, Bifurcations of one and two dimensional maps, Phil. Trans. Soc. London A 311 (1984) 43-102. [25] P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birldaauser, Boston, 1980). [26] B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphlsms on the real axis and representation of numbers, Ann. Inst. H. Poincar6 A 29 (1978) 305-356. [27] L. Jonker and D.A. Rand, Bifurcations in one dimension I: The nonwandering set; II: A versal mode for bifurcations, Invent. Math. 62 (1981) 347-365; 63 (1981) 1-15. [28] G. Burde and H. Zieschang, Knots (De Gruyter, Berlin, 1985). [29] R. Bowen, On Axiom A Diffeomorphisms, CBMS Regional Conference Series in Mathematics 35 (AMS Publications, Providence RI, 1978). [30] J. Moser, Stable and Random Motions in Dynamical Systems (Princeton Univ. Press, Princeton, N J, 1972), [31] J. Guckenheimer, On the bifurcation of maps of the interval, Invent. Math. 39 (1977) 165-178. [32] J. Guckenheimer, Sensitive dependence on initial conditions for one dimensional maps, Comm. Math. Phys. 70 (1979) 133-160. [33] J. Guckertheimer, Bifurcations of Dynamical Systems, in: Dynamical Systems, J,K. Moser, ed., (Birldaauser, Boston, 1980). [34] D. Singer, Stable orbits and bifurcations of maps of the internal, SIAM J. Appl, Math, 35 (1978) 260-267. [35] J. Guckenheimer, The bifurcation of quadratic functions, Ann. NY Acad, Sci. 316 (1979) 78-85. [36] R. Devaney and Z. Nitecki, Shift automorphisms in the Htnon mapping, Comm. Math. Phys. 67 (1979) 137-148. [37] M. Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory I, Applied Mathematical Sciences, Vol, 51 (Springer, Berlin, 1984). [38] M. Golubitsky, I. Stewart and D.G. Sehaeffer, Singularities and Groups in Bifurcation Theory II, Applied Mathematical Sciences, Vol. 69 (Springer, Berlin, 1988). [39] N.K. Gavrilov and L.P. Silnikov, On three dimensional dynamical systems close to systems with a structurally unstable homoclinic curve, I, Math. USSR Sbomik 88 (4) (1972) 467-485. [40] N.K. Gavrilov and L.P. Silnikov, On three dimensional dynamical systems close to systems with a structurally unstable homoclinic curve, II, Math. USSR Sbomik 90 (1) (1973) 139-156.