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Ordered orbits of the shift, square roots, and the devil's staircase

Shaun Bullett and Pierrette Sentenac

Mathematical Proceedings of the Cambridge Philosophical Society / Volume 115 / Issue 03 / May 1994, pp 451 481DOI: 10.1017/S0305004100072236, Published online: 24 October 2008

Link to this article: http://journals.cambridge.org/abstract_S0305004100072236

How to cite this article:Shaun Bullett and Pierrette Sentenac (1994). Ordered orbits of the shift, square roots, and the devil's staircase. Mathematical Proceedings of the Cambridge Philosophical Society, 115, pp 451481 doi:10.1017/S0305004100072236

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Math. Proc. Camb. Phil. Soc. (1994), 115, 451 4 5 1

Printed in Great Britain

Ordered orbits of the shift, square roots, and the devil's staircase

B Y SHAUN BULLETT

School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road,London Ei 4NS

AND PIERRETTE SENTENAC

Mathe'matique, Bdtiment 425, Universite de Paris-Sud, 91405 Orsay, France

(Received 25 January 1993; revised 3 June 1993)

Abstract

An orbit of the shift <r: t*-*2t on the circle T = R/Z is ordered if and only if it iscontained in a semi-circle C =\ji,/i + | ] . We investigate the 'devil's staircase'associating to each / teT the rotation number v of the unique minimal closed o~-invariant set contained in C ; we present algorithms for fi in terms of v, and we prove(after Douady) that if v is irrational then ji is transcendental. We apply some of thisanalysis to questions concerning the square root map, and mode-locking for familiesof circle maps, we generalize our algorithms to orbits of the shift having 'sequencesof rotation numbers', and we conclude with a characterization of all orders of pointsaround T realizable by orbits of cr.

Resume

Une orbite du shift a: t^-2t sur le cercle T = [R/Z est ordonnee si et seulement sielle est contenue dans un demi-cercle C = \ji,/i + \\. Pur chaque fisT, il existe ununique sou-ensemble minimal ferme cr-invariant contenu dans Gp ayant un nombrede rotation v; la correspondance (/*, v) est un ' escalier du diable'; nous donnons unalgorithme pour calculer /i en fonction de v et nous prouvons (apres Douady) que siv est irrationnel, alors /i est transcendant. Nous donnons une application de cetteanalyse a l'etude de fonctions 'racine carree et racine A-ieme' sur le cercle.

Nous generalisons l'algorithme a l'etude d'orbites du shift ayant une chaine denombres de rotation, et caracterisons les ensembles de points ordonnes sur T,realisables par une orbite de o~.

Introduction

We consider orbits of the 'doubling map' o~: t^-2t (mod 1) on the circle U./Z (the'squaring map' when we think of (R/Z as the unit circle in the complex plane).A closed subset A of U/Z is described as ordered under a it A is invariant (that is,o-(A) = A) and if a preserves the cyclic order of the points of A. Such a subset has arotation number, defined as the rotation number of any degree 1 order-preserving self-map of [R/Z extending cr on A, or, equivalently, as the frequency with which the digit' 1' occurs in the binary expansion of any chosen point teA (see Part 1: Section 1).This paper is concerned with the complete classification of such subsets A, explicit

452 S H A U N B U L L E T T AND P I E R R E T T E SENTBNAC

I

Fig. 1.

algorithms for their construction, some of their number-theoretic properties, ageneralization of the notion of order, and a characterization of all orders of pointsaround IR/Z which are realizable as orbits of cr.

We show in Section 1 that a closed subset A of IR/Z, invariant under cr, is orderedif and only if it is contained in some closed interval of length 1/2. It follows that onepractical method to find such ordered sets is to iterate the ' halving' (or ' square root')map cr"1 :t^t/2 (mod 1), restricted to any fixed interval lg = [6/2, (6+ l)/2) c IR/Zof length 1/2, starting the iteration at an arbitrary initial point t0. As is evident froma simple computer experiment (and as will be proved later in this article) any orbitof cr"1 constructed in this manner converges rapidly to an attractor Ag which dependsonly on 6 and not on the initial point chosen, and which moreover is ordered underthe shift cr.

Example. For any 6e (1/3,2/3) the attractor Ae is the periodic orbit {u0, uj whereu0 = 0-01 and Wj = 0-10 (in binary, and where the bar denotes an infinitely repeatedsequence of identical blocks). For 6 = 1/3, while the set {u0, %} still lies in the intervalIg it is no longer fully-invariant under cr"1, as u0 has an alternative image, other

1 / 3,than u1; in

is K . i i J UAm = {uo,Ul} U

namery the point t>1 = 0-0(01)(= 1/6). The full attractor A1/3w h e r e ^ = 0-(10)J'0(10), w2;._1 = 0-(01):/-10(0T). Similarly

where w2} = O(01)' 1(01), wv_x =11(10).

We prove that the attractors Ag for the restrictions of cr'1 to the intervals Ig arethe unique closed subsets of IR/Z which are invariant under the shift a and whichhave C3rclic order preserved by a. Furthermore we show that if Ag has rationalrotation number p/q then Ag is one of three sets A^'9, AZlq, Aplq which depend onlyon p/q, and that if Ae has irrational rotation number v then Ag = A", a Cantor setwhich depends only on v. The three possibilities with rotation number 1/2 are (as

Ordered orbits of the shift 453

+-+-•

Fig. 2.

indicated by the example above) A1!2 = A1/3, A]!2 = A2/3 and A1/2 = Ag for any1/3 < 6 < 2/3. The first two are maximal and the third (their intersection) minimal.We give explicit algorithms for the construction of the sets AVJ9 and A" for bothrational and irrational v, including algorithms based on continued fractionexpansions.

The graph which assigns to each Ig the rotation number (under a) of the set Ao isa devil's staircase (Fig. 1), that is to say a continuous weakly monotonic map of theinterval to itself which is locally constant on a set of full measure without beingglobally constant. The horizontal steps on this graph (which together occupy fullmeasure) correspond to rational rotation numbers p/q. At the left-hand end of thestep for p/q the attractor Ae for cr"1 is AZl9, at the right-hand end it is Afla, and inbetween it is Avlq,

An identical graph appears in the combinatorial theory of the Mandelbrot set [6,8, 9]. It arises in the following way. The Mandelbrot set is defined to be

M = {ceC: the orbit of 0 underfc: zi^z2 + c is bounded}.

See Fig. 2. Douady and Hubbard[8, 9] showed that the complement of M in theRiemann sphere C is conformally equivalent to the disc C — D (where D denotes theclosed unit disc) via:

(where care is taken to choose the appropriate 2nth root in this expression). Theinverse image under <t> of the line {z: arg (2) = 8} is known as the external ray of

454 S H A U N B U L L E T T AND P I E R B E T T E SENTENAC

argument 6 e IR/Z in C —M. The cardioidi>f0, the main component of the interior ofilf,is also conformally equivalent to a disc [8, 9]. Mo consists of the values of c where /„has an attractive fixed point other than oo, and it is parametrized by the derivativeof fc at that fixed point. This parametrization defines the internal rays in Mo. Eachinternal ray of rational argument has end point on dM0 which is also the end pointof two external rays. The devil's staircase (Fig. 1) appears as the assignment ofinternal to external arguments along dM0. The underlying reason for this is asfollows. When ceM the nlled-in Julia set Kc of/c has complement parametrized by

(as shown in [8, 9]). This parametrization defines the dynamic rays in C—Kc. Notethat fc acts on the arguments of these rays as the binary shift. When fc has arationally indifferent fixed point, say of derivative e2niplQ, Kc is a 'flower' with q'petals', separated by q dynamic rays. The map fc permutes these dynamic rayscyclically, with rotation number p/q, and the analysis [6, 8, 9] of this action and thatof the Douady—Hubbard map O yields Fig. 1 as the graph of the assignment ofinternal to external arguments.

Yet another appearance of the same combinatorics is in the study of kneadingsequences of Lorenz maps of the interval [12, 15]. In these articles, Glendinning,Sparrow and Hubbard consider pairs (a,/?) of binary numbers, 0 < a < /? < 1,satisfying the condition

a ^ an(oc) <P, a < <rn{P) ^ /?, Vn ^ 0. (*)

These they call allowed pairs. I t is shown in [15] that (*) is both necessary andsufficient for (a, /?) to be realizable as the kneading invariant of an expanding Lorenzmap. But the allowed pairs of form (a,a+1/2) are easily seen to be the pairs ofminima and maxima of our closed ordered sets A", with v irrational. It follows, forexample, that the points on the 'diagonal' (/? = a+1/2) of figure 3 of [15],correspond to precisely the points on the devil's staircase (our Fig. 1) where therotation number is irrational. (See also Remark 2 at the end of Part 1, Section 3,where we discuss the relationship between 'order' and 'trivial renormalizability'.)

The ordered sets considered in this paper have been studied in one guise or anothersince the earliest days of symbolic dynamics. Hedlund and Morse [17] investigatedSturmian trajectories, namely sequences (infinite or bi-infinite) in two symbols a andb with the property that the numbers of bs in any two segments of the same lengthdiffer by at most one. Their basic examples of such sequences were given by thefollowing construction:

Let a be a positive real number, let (S = I/a, and let c be an arbitrary real number. Onthe real axis — oo < x < oo consider the set of points

Let T(c,a) (respectively T'(c,a)), denote the sequence ... abj<abj<+i... where j i denotes thenumber of points of (*) in the interval i ^ x < i +1 (respectively i < x ^ i + 1). For arational and c = m integer let S(m, a) (respectively S'(m, a)) be as above except that thesingle value j m is now to be the number of points of (*) in the interval m ^ x ^ m+ 1(respectively m < x < m+ 1).


Hedlund and Morse called these sequences T(c,a), T'(c,oc), S(m,a), S'(m,a)mechanical sequences. A major result of their paper [17] is that all Sturmian sequencesare of the three types: irrational (T(c,a) or T'(c,a) for a irrational), skeiv (S(m,a) orS'(m,a) for a rational) or periodic (T(c,a.) for a rational). These three typescorrespond to our A" (v irrational), AVJ9 (p/q rational) and Avlq (p/q rational)respectively: see Part 1: Section 4. The mechanical construction of Hedlund andMorse is our staircase algorithm (Theorem 3).

Gambaudo et al. [11] in the case of v rational, and Veerman[19, 20] in the case ofv irrational, proved that when the trajectories are written with 0s and Is rather thanas and 6s, interpreted as binary expansions of real numbers (mod 1), and regarded asorbits of the shift <J on 1FS/Z, the Sturmian condition (referred to as the 'optimal'condition by Veerman) is equivalent to the preservation of cyclic order.

Motivated by the combinatorics of the Mandelbrot set, our approach to the studyof orbits of o~ starts from the question of preservation of cyclic order, rather thanfrom the Sturmian condition favoured by these earlier authors. Our initial (almosttrivial) observation is that an orbit has its order preserved by a if and only if it iscontained in a closed interval of length ^1 /2 (Proposition 1, first proved byGambaudo et al. [11] for v rational, and Veerman [19, 20] for v irrational, both byindirect methods). We give a complete classification of ordered a-invariant closedsets and of the intervals of length 1/2 containing each of them (Theorems 1 and 2).Theorem 1 repeats the results of Hedlund and Morse, but in the language of orderedsets. Much of the content of both of Theorems 1 and 2 is explicit or implicit in the workof Veerman [19, 20], and related results, in the context of the Mandelbrot set, havealso been obtained recently by Atela[2] and by Bandt and Keller [3, 4, 5]. However,as the same symbolic dynamics arises in many different contexts and the literatureis somewhat scattered and imprecise, we believe it to be a worthwhile exercise topresent a unified purely combinatorial treatment, and in particular to classifyexplicitly the ordered closed invariant sets. These include the skew Sturmiansequences A ̂ !9, which play a key role in marking the end points of steps on the devil'sstaircase.

As we have already observed, Theorem 3 (the staircase algorithm) is closely relatedto the mechanical construction of Hedlund and Morse, but as a corollary we are ableto present in Theorem 4 explicit algorithms for the construction of dv^q (the endpoints of the p/q step on the devil's staircase) based on the continued fractionexpansions of p/q. As far as we know these are new. In the limit for the sequence ofcontinued fraction convergents of an irrational v they yield the same expression for6" as was first obtained by Veerman[19] (using what he termed 'scaling sequences').As corollaries of our algorithms we obtain two further new results, first an explicitalgorithm for the continued fractions of d\ (y rational) and 6" (v irrational) in termsof the continued fraction expansion of v, and secondly, a new theorem of Douady,which he discovered while we were preparing this article, that if v is irrational then6" is transcendental.

In Part 2 of the paper we look at applications and generalizations. We begin byinterpreting our results for a in terms of the square root map o~~x, in particularjustifying the observations that we made in the second paragraph of thisIntroduction. We then extend our consideration to orbits of a on U/Z which are notordered. We identify a class of orbits which have chains of rotation numbers: these

456 S H A U N BTJLLETT AND P I E R R E T T E SENTENAC

are obtained by using as building blocks periodic sequences for different d^"19". Themethod is closely related to that of Theorem 4 (and to renormalization, that is,Douady tuning, in the terminology of the Mandelbrot set). Our final result, Theorem8, is a complete characterization of all the cyclic orders of points that can be realizedby orbits of a.

I t is in the nature of this subject that many of the results have been discovered andrediscovered at different times by different authors from different points of view. Wegive references just before the statements of our theorems, wherever we know ofrelevant earlier work. Where we give no references we believe our results to be new.We apologize for any attributions we have omitted through ignorance.

Part 1. Orbits of the shift a

1. Introduction

Let A be a subset of the circle T = R/Z and / be a map A -> T (not necessarilycontinuous). We say t h a t / i s order-preserving if for each triple (a, b,c) in A the triple(f(a),f(b),f(c)) lies in the same cyclic order around T, or else is degenerate (i.e. two,or all three, of the points coincide).

LEMMA 1. Every order-preserving map f:A^T, defined on a subset AofT, extendsto an order-preserving map F: T -> T.

Proof. Let n denote the projection R-s-IR/Z. Choose a 'base point' aeA andrepresentatives a,beR for a and b=f(a) (that is, n(a) = a and n(b) = b). Nowdefine F(d) = b, F(d+l) = b+l, and extend I to a (weakly) monotonic map[a, d + 1] ->• [b, b + 1]. There may be many ways to make such an extension: we givean explicit one in detail as we shall use a similar construction much later, in theproof of Theorem 8.

(i) if x = n(x)eA and/(x) =t= b then define F(x) to be the unique representative off(x) in ]b, 6 + l [ ; if xeA (x #= a) has/(a;) = b define F{x) to be b + l if there is a yeAwith x > y &ndf(y) #= b, else set F(x) = b;

(ii) if xeJ—A is a limit of points xteA with xi<x (in [a,d+l[) then setF(x) = \imF(xt) (defined by (i));

(iii) if xeJ— A is not a limit of the form in (ii), but is a limit of points xteA withxt > x, then again set F(x) = lim-F(iy;

(iv) sinceF(x) has now been denned for all xeA, by (i), (ii) and (iii), we may extendF to the whole of [a, a + 1] by linear extrapolation.

Finally, we extend F to a map F: M.-+M by periodicity, that is by sending each[d + n,d + n+l] to [b + n,b + n+l] in the same way. It follows from the order-preservation of/ on A that F is (weakly) monotonic increasing and hence induces anorder-preserving map F: T-s*T. I

Any order-preserving map of the circle F: T -> T has a well-defined rotationnumber, the fractional part of pn

lim ——n-*co n

for any lift/ : U-+M satisfying F(y+l) =F(y)+l for all ye R, and for any xeU.


9/31

10/31 5/31

Fig. 3.

Rigid rotation

While the extension F of / constructed in Lemma 1 is far from unique in general,iif(A) cz A then the rotation number of F depends only o n / (since we may choose xto be a lift of a point in A).

We now specialize to the case when/ = o~A, the shift a restricted to a subset A ofTwith a(A)=A.

PROPOSITION 1. The map o~A is order-preserving on A if and only if A is a subset ofa closed semicircle.

Proof. Any triple contained in a closed semicircle has its order preserved by <r,since a doubles length. Conversely, if (a, b, c) are not contained in a closed semicircle,then (a,6+1/2,c) are, and hence (a(a),a(b),cr(c)) = (cr(a), a(b+ 1/2), a(c)) has thesame cyclic order as (a, 6+ 1/2, c), namely the opposite to that of (a,b,c). I

It is easy to describe all the possibilities for the subsets A of a semicircle whichsatisfy a(A) = A and contain 0:

A = {0}, A = {0} U {l/2n}n > 0, A = {0} U {1 - l/2n}n>0.

In each case the rotation number vA is equal to 0. These are the invariant sets A0, A\and At in the notation of Theorem 1 below.

Remark. A way to compute vA: Let A be contained in a semi-circle, andcr(A) c A. For xoeA. let xn = an(x0), n ^ 0, and let xQeU be the unique lift ofx0 with 0 ^ x 0 < l . Inductively, let xneU be the unique lift of xn such that0^xn-xn_1 < 1. Then

„ =

Moreover, if we represent a;0 as a binary number, then xn is obtained from xn_1 by:(i) Shift the fractional part one binary place to the left;

(ii) Add the overflow (0 to 1) to the integer part.

It follows that the integer part of xn is the number of Is among the first n digits

458 S H A U N B U L L E T T AND P I E R R E T T E SENTENAC

of the binary expression of x0 and that vA is the frequency with which the digit ' 1'occurs in this expression.

Example.4 /_5_ io 20 i . 18\ 4 t— rJL J L j - I l-" ~ 131 ' 31 > 31 ' 31 J 31/ ' ** L31 ' 31 ' 2-1"

*0 = 00101, ^ = f .

An alternative characterization of vA is to say that the points of any orbit c Aoccur in the same cyclic order, up to degeneracy, as those for a rigid rotation throughvA (Fig. 3).

2. Results of Part 1

In the following, we consider subsets A of T which are closed and tr-invariant(aA =A). We ask:

1. Given ve R/Z, does there exist such a subset A" with rotation number vi If so,is it unique ? What relationship is there between v and the set of values of fi such thatthe semicircle CM = [/*,/*+ f] contains Av%

2. Given /ie]0,%[, is there a subset A of this type contained in the semicircleC^ = \JX,/I + J\ ? If so, is it unique? What relationship is there between /i and therotation number v of AM ?

Theorems 1 and 2 answer these questions. Theorems 3 and 4 provide further detailsof the correspondence between fi and v (the devil's staircase), in particular algorithmsfor A" and properties of the numbers involved. Theorem 1 parts (a) and (c) areessentially a restatement of results of [17] in the language of ordered sets, and both(a) and (b) are also proved in [11, 19, 20]. Theorem 2 can be deduced from the resultsof [11, 19, 20], and related results to Theorem 3 can also be found in these articles.However, since it is not easy to extract the statements of Theorems 1, 2 and 3 inprecisely the form below from these sources, and since several of our methods andconstructions are new, we give complete self-contained proofs. Theorem 4 is new.

THEOREM 1. (a) For each v e IR/Z there exists a unique minimal closed a-invariantsubset A" cz T with rotation number v.

(i) If veQ/Z, A" is finite and consists of a single periodic cr-orbit.(ii) If ve (IR — <Q)/Z, A" is a Cantor set and consists of the closure of the cr-orbit of a

recurrent point.(b) For each v, the set A" has zero measure and zero Hausdorff dimension.(c) For each ceQ/Z, there exist two maximal closed a-invariant sets Av_ and A\ with

rotation number v, and furthermore A" = A"_ f) A\.For ve(U — Q)/Z, A" is the unique closed a-invariant set with rotation number v.

THEOREM 2. (a) For each /ie]0,|[, the semi-circle C^ = \ji,/i + \] contains a uniqueminimal closed a-invariant set A .

(b) The graph of the rotation number v ofA^, as a function of fi, is a devil's staircase.In particular v is rational except on a set of values of ju which has closure of measure zeroand Hausdorff dimension zero.

By a devil's staircase we mean the graph of a function which is continuous andmonotonic, and which is locally constant on a set of full measure without beingglobally constant.


3 4 5 66f/2= 0.010101...

3 4 5 66>,+/2 = 0.1010...

Fig. 4.

THEOREM 3 (Algorithm for A"). (a) For each v 4= OeQ/Z, there exists a unique pairof rationals 0 < 6~ < 0+ < 1 [constructed algorithmically), such that:

(i) 0~ and 0+ are adjacent points on the a-orbit A";(ii) 0+/2 and 0J/2 +f lie on A", and A" c [0+/2,0;/2 +f]. / / v = p/q in reduced form,

the interval [0+/2,0;/2+i] Aas ZercgrtA (29"1- l ) / (2«- l ) .(6) .For eacA ve(IR —Q)/Z <Aere erases a unique real number 0 < #„ < 1 swcA <Aa£:

(i) 0,, lies on A" and its a-orbit is dense in A"; (ii) 0,,/2 and 6J2 + \ lie on A", and

(c) / / v0 < v1 < v2, with vleQ/Z and v0>v2e{U-Q)/Z, then 6^ < 0" < 0+ < 0r2.

The numbers 0± (veQ/Z) and 6V (ve(U — Q)/Z) are constructed by explicitalgorithms. That for 6^jq is given by drawing a line of slope p/q through the originin 1R2, and then reading off the binary expressions for dijj/Q from two 'staircases' ofinteger lattice points, one touching the line and the other just below it (Fig. 4).Starting from the point 1 on the horizontal axis we write ' 0' for a horizontal stepfollowed by another horizontal step, and ' 1' for a horizontal step followed by avertical one.

More details are given in Proposition 3 (the 'staircase algorithm'). The algorithmfor dv, ve(M. — Q)/Z, is similar (Proposition 4), and a formulation based on thecontinued fraction expansion of v is given in Proposition 5 and its corollaries.

The next result concerns number-theoretic properties of 6V for irrational v. Recallthat an irrational number x is said to be fl-Diophantine if there exists a positive realnumber c such that for all rational p/q, \x—p/q\ ^ c/q? and Liouville if it is not/?-Diophantine for any /?. Roth's Theorem [18] states that if x is algebraic over Qthen x is /?-Diophantine for all /? > 2.

THEOREM 4 (Transcendence ofdv,ve(U — Q)/Z). (a)Ifve(U — <Q)/Z is not of constanttype, then dv is Liouville.

(b) If ue(U — Q)/Z. is of constant type, then: (i) if v is not noble, then dv is not3-Diophantine; (ii) if v is noble, then 6V is not fl-Diophantine for /? = 2 + y — e, ivherey = 0-618... is the golden mean and e is any positive real number.

Hence, by Roth's theorem, dv is transcendental for all j>e(IR — <Q)/Z.

Recall that v is said to be of constant type if the sequence of integers (a1,a2,a3,...)in the continued fraction expansion v = l/(a1 + l/(a2+ ...)) is bounded. and it is said tobe noble \ian= 1 for all sufficiently large n. For example the golden mean y, whichhas an = 1 for all n, is noble.

460 SHAUN BULLETT AND PIERRETTE SENTENAC

We thank Adrien Douady for Theorem 4, the statement and proof of which heoutlined to us while this article was being prepared.

3. Propositions and lemmas for the proof of the theorems

Notation. Consider the binary expression {6j(x)}j>1 and the sequence of integersj(x)}jî defined for each x e T by

CO - l~.\ 00 „ /™\

r _ iF tT\ F (T\ p iT\ \ _ v j( > — y W *

with e}(x)e{0,1}, Pj(x) = ex(x) + ... +Sj(x) (if x has two binary expressions, choose theone with an infinity of zeros).

We note that the pre-images of x under a are

x x 1-={0,el(x),...,ei{x),...) a n d - + - = ( i , e t ( x ) , . . . , e } { x ) , . . . ) .

Consider the points {j'Pj(.x))j-»i ' n "̂ 2 a n d define the staircase of x by:(i) ttPj+Ax) = PM) (ei+i(x) = ° )> t h e Points {j,Pj(x)) and (j+i,pj+1(x)) are joined

by a horizontal line;(ii) if pi+1{x) = p}(x) + l, the points are joined by a horizontal line followed by a

vertical line.Conversely, every sequence of integers (jO;)3>1, such that 0 ^ Pj+\~Pj ^ 1 f°r a ui>

provides a number x e T (Fig. 5).

PROPOSITION 2. Let xeT and let 0x denote the forward orbit of x.(i) The orbit 0x is contained in the semi-circle Cx/2 = [x/2,x/2 + \] if and only if the

following condition is satisfied:

(ii) / / (C) is satisfied, then a\o has rotation number vx = limÔDp;(a;/2)/j and thestaircase {pj(x/2)} satisfies

( | ) ^ ( | ) 1 . (C)

Proof, (i) S u p p o s e 0x cz Cx/2. Let

w h e r e x = {elte2, ...,ep ...), so x/2 = ( 0 : e 1 : e 2 , . . . ) .

We claim that

The proof is a consequence of an obvious remark. For every ke{0,1,2,...} a wordof length k, 'et ...et ' cannot occur in the binary expression of x/2 (or x/2 +1) with


1 2 3 4 5 6 7Fig. 5. The staircase of \ = (JT.

two zeros at its extremities on one occasion and two ones at its extremities onanother occasion, since then the points

could not lie in the same semicircle, as the difference between the middle two wouldbe ^ \. For instance, for k = 0, the rule is that the orbit cannot contain both '00' and'11 ' .

We shall prove by induction on I that, for every JceN, 0 ^ sk t ^ 1. In fact ourinductive hypothesis is that 0 < sfcj j < 1 and

(or = 1

if sk , = 1, then { or

For 1=1, we havesk 1 = ek s o 0

but x2'

(P)

462 SHAUN B U L L E T T AND P I E R R E T T E SENTENAC

So, using the fact that x/2 ^ crk(x/2) ^ x/2 + \, we are done.Now suppose that (P) is true for 1 ^ j < I, so that 0 < sk<l_1 ^ 1 and

or

= e,

_ i

*,* = 1.

or

= 0, el =

Now if e = e,7+1 we are done, and if elc+l+1 4= e m we also get the required result,using the remark above concerning sub-words surrounded by a pair of 0s or Is.

Conversely, suppose that Ox satisfies (C). We must show that Ox c [x/2,x/2+\].Let ak(x/2) be the first point of the orbit lying outside this range, and consider

x2

x 1

Let I be minimal such that efc+i =t= et. Then, since ak {x/2) lies outside [x/2, x/2 + | ] ,if efc = 0, we have e( = 1 and ek+l = 0, and if efc = 1, we have et = 0 and ek+l = 1.

These give sk l+1 = — 1 and 2 respectively, contradicting (C).(ii) Immediate from the definition of rotation number. I

PROPOSITION 3. Let p/q be a (reduced) rational number, 0 <p/q < 1.(i) There exist exactly two real numbers, 0 < d~/g < 6p/g < 1 (both rational) such that

dp/g/2 satisfy condition (C) and the orbits of 6^/g both have rotation number p/q.(ii) The a-orbits of 6~/g and d*/g coincide, are q-periodic and are contained in the

interval [61J2,(6-lq+\)/2}.(iii) ?p/9 has an even numerator and 6p/g has an odd one.

Proof. Given 0 < p/q < 1, condition (C) of Proposition 2 defines two 'extreme'staircases (as in Fig. 4). The first is where p}(x/2) always takes the larger valuewhenever there is a choice, and the second is where it always takes the smaller value.Denote these two staircases by (j,pf) and (j,pj) respectively. Explicitly we definethe staircase algorithm by

£>+ = pj = integer part ofjp/q if j is not a multiple of q;

p+ = kp and pj = kp—1 if kq =j.

The following elementary properties are self-evident from a consideration of the twoextreme staircases.

(I) (a) VJfc,


(b) e}=p]-+1-pf = e++], Vj, e+_! = 1, ej = 0,e7 = e-+;-, Vj, e-_! = 0, e" = 1,

(c) ef = ej = e}, j = 1,2, ...,q-2,

(II) (a) 3 j 1 e{ l ,2 , . . . ,g - l} such that j^p/q-p^ = 1/q,3j 2 e{ l ,2 , . . . ,g - l} such that j2p/q-p£ = 1-1/q, J i + j 2 = ?=(b)

We define 0*/9 and #~/9 to be the binary numbers associated to the staircases'Pf) a r )d (j,Pj) respectively, that is to say (by (I) above):

ft+ = IF F F 1VlQ \ °1 ? ° 2 ' " ' ' ' ff—2-' '

Vvia = (^1 > ^ 2 ' • • • ' ô—2' ">

where the bar indicates a repeated block of symbols.(i) By (II) above, 6^/g satisfy (C). We have to check that no other sequences do.

Since (C)=>(C) by Proposition 2, it suffices to check the 'intermediate' staircasesdenned by (C). For such a staircase either there exist k,l (possibly equal) with

or there exist k, I with

and in either case condition (C) fails.(ii) <r>*e-plq = 6+lt a n d o-W%lq = d-lq from ( I I ) .

(iii) This follows at once from the explicit expressions for 6~/g and 0£/9 above. I

An example of Proposition 3. Take v = 2/5 (Fig. 6).

- y = (0,0,1,0,1) = &, 0t,s = (0-1.0,1,0)

= (0 ,0 ,1 ,0 ,0 ,1) = 4 0J/B = ( o , l . o . o , i ) =

10 20 9 181 f- r 5 20T,3i, 31, 51,31/ ^ L3T'3lJ-

PROPOSITION 4. Let v = M. — Q, 0 < v < 1. (i) TAere is a unique real number0 < dv < 1 5MCA <M< 0,,/2 satisfies condition (C) cmd <Ae or6i/ o/ 5W has rotation number v.

(ii) The cr-orbit of 6V is contained in the semicircle [6J2,6J2 + \].(iii) dv is recurrent under a.

Proof. Condition (C) this times defines a unique staircase, since 0 < jv—pj < 1 => p}

is the integer part oijv. This staircase satisfies condition (C) of Proposition 2, so weat once have (i) and (ii).

To show that 6V is recurrent, given keN set

mk = max (jv — Pj), with 0 < mk < 1.


ei,/5 = 0101015

Fig. 6.

The sequence {jv—pî>x is dense in [0,1], so there exists I such that 0 < lv—ph <

For j = 1,2,...,A;+1, the inequalities

0 <jv—pi < mk,

imply that pl+j = pt + pr

For j = 1,2,..., k, we deduce that el+i = e^ and hence that the binary expressionsfor <Jldv and for dv coincide for their first k digits, completing the proof. I

In Proposition 5, we give another algorithm for &*, this time utilizing thecontinued fraction expansion of v. We first note the following consequence of thestaircase algorithm.

LEMMA 2. Given a/b, a'/b' (reduced) rationals which are adjacent (a'b — ab' = 1), thebinary expressions of d~,/b. and d+/b coincide for their first b + b' — 2 places.

Proof. This follows directly from the staircase algorithm. Note that 8~,/b, (resp.da/b) have periodic binary expression of period b' (resp. b) and that

toGiven a sequence (finite or infinite) of positive integers (alta2,...), definebe associated sequence of rationals

Pi 1 1

a

If ai+1 * 0,

3

?m = 13

Pm = P

1 '1 a2

i-i+a-j+iP

• • • s

••> q<>

'P Po

1)

= 1,= 0,

% +

P i

a2 +

= «i

= 1.

1

...I/a,

It is well-known that two consecutive fractions in such a sequence are adjacent,and that the sequence splits into two monotonic subsequences


PROPOSITION 5. (i) The binary expressions for d~ /g (j odd) and d^ /Q (j even) maybe obtained as folloivs:

0^01=^ soa, digits

<%,/«,,= M i - d i O so 0 + ^ = ^ ^ where fc8 = 2(l + 2«.a2copies

Inductively, d~ /g (j odd) and 6* /g (j even) are defined by dp where

and V./««-Example. How to compute

2 _ 1 1__ _ _

2 +1 + 1/1^ = 0

(2,2)- . ( ^ , = 01

16*2/5 = 01001.Proof of Proposition 5.(i) By Lemma 1, Q\la =d1 = (00... 01); then apply Lemma 2 and induction.

a,

(ii) F o r j = l

*;,/«, - ^ « , = d,-d2, d2 = a^d, 0

7 1 j _ 2«̂ ^ 1 29«-21 ~ 2 « > - 1 ' 2 ~ 2«»-l^_12«^~ (2«>

Hence d1 — d2= l/((29' — 1) (292— 1)) and the general result follows by induction. I

COROLLARY 1 (how to compute 6f for v = p/q). For rational v the continued fractionhas finite sequence (a1,a2, ...,ar = 1).

(a) TC . \&l = &t . and apply Proposition 5,If r is even < Pr/Qr x

466 S H A U N B U L L E T T AND P I E R R E T T E SENTENAC

(b) I f r .g o d d \er = d~r/Qr and apply Proposition 5,

to = 2*-<wCOROLLARY 2 (how to compute 6,, for ve(U — Q)/Z).IfveM — Q, so that the continued

fraction has an infinite sequence of positive integers, then

(i) ^ < ^ < ^ + 1 / W V ^ ° >and 6V is the common limit of the two sequences.

(ii) The sequence (A1,A2, ...,Ap ...) of positive integers in the continued fractionexpression for 0V is given inductively by

Remark 1. Our construction (above) of ordered orbits of the shift translates intoalgorithmic form the set of equivalent conditions proposed in [11] in the case ofrational rotation numbers. Moreover our algorithms enable us to construct themaximum and minimum points of all closed ordered sets, whether the rotationnumber be rational or irrational.

Remark 2. A sequence £ of Os and Is is said to be trivially renormalizable [12, 15]if either

0£ = w_ w™1 w™* w™3...,

where w_ = 0 and w+ = 10 or w_ = 01 and w+ = 1, and 0 < mt ^ oo, or

with w_ and w+ as above and 0 < nt ^ oo.The renormalization of £, is v where

in the respective cases. For sequences constructed by the staircase algorithm(Proposition 3) trivial renormalization corresponds to the action of generators of themodular group PSL(2, Z) on the underlying integer lattice. (This observation is whatlies behind the 'reduction' method of [11] and the 'scaling sequences' method of[19].) It can be deduced that g is trivially renormalizable infinitely often if and onlyif £ is of the form d+/2 or 0J/2 + 1/2 for v rational, or 6J2 or 0J2 + 1/2 for virrational, that is, if and only if £ is an end point of one of our closed ordered sets A".Trivial renormalizations can also be used to produce alternative constructivealgorithms for 0^ and 0V.

4. Proofs of theorems

Proof of Theorem 1. (a) (i) If veQ/Z, then by Proposition 3 the cr-orbit of &*• isfinite (closed), periodic and therefore minimal, and has rotation number v. Hence itprovides an A".


(ii) If ce((R —<Q)/Z, then let A" be the closure of the cr-orbit of the recurrent point8V of Proposition 4. It is clearly a closed, <r-invariant minimal subset with rotationnumber v (notice that a(A") = A"; each point has one pre-image except $„). The orbitof dv is dense in A", and A" contains no interior point, so A" is a Cantor set.

It remains to show uniqueness, in both cases. Since A" has a rotation number, itis contained in some semicircle, and since A" is closed we can speak of its least pointx/2. Then xeA" (since A is cr-invariant) and 0x £ [x/2,x/2 + \\. Uniqueness nowfollows from Propositions 2, 3 and 4.

(6) For veQ/Z, the set A" is finite and there is nothing to prove. For pe(U — Q)/Z,a is injective on A" except at the points 6J2,6J2+\. Remove the o-~1-orbit of 8J2and consider Av = A" — {\Jn^Qa~n(6l,/2)}. Then a is a continuous bijection of Av toitself. But er doubles measures. It follows that m(Av) = 0 and hence m(A") = 0 (sinceAv is obtained by adjoining countably many new points to A"). The same argumentproves that the rf-dimensional measure md{Av) = 0 for all d > 0, and hence that Av hasHausdorff dimension zero.

(c) For fie]8^/2,8^/2[, the only invariant subset of C^= [/t,/t + |] is A", but for/i = d~/2 we can adjoin to A" the point d~/2 and all its inverse images lying in Cr

This gives a maximal set we denote A"_. Similarly for /* = 6^/2 we can adjoin toA" the point 6+/2 + \ and all its inverse images in C^, to obtain A\. In the case ofirrational v, since 8J2 and 8J2 + \ both lie in A", this set is maximal as well asminimal. I

Example. For v = \,Av_ consists of all points of the form 0-00100100 ... 0010001001and images under o~ (i.e. we insert at most one extra 0 into an expression for a pointof A"), and A\ consists of all points of the form 0-001001... OOIOIOOT.

Proof of'Theorem 3. (a) is immediate from Proposition 3.(b) is immediate from Proposition 4; note that uniqueness of 8V ensures that both

8J2 and 8J2 + \ lie in A", else the ' least' and ' greatest' elements of A" would give new

(c) is immediate from Proposition 3.

Proof of Theorem 2. Since every semicircle C^ = \ji,/i + \\, /i€~\0,\[ contains someA", we have

]0,|[= U /,u U {-),

where /„ = [8~/2,8+/2]. To prove that the /„ occupy full measure we observe that byProposition 3

•̂ -J "l'\Lvl Ct S—l O9 1 '

where the sum on the right-hand side is taken over all p/q in reduced form. Thecalculation of this sum is standard. We give an elementary proof. Consider the subset2̂ = {(p,q)\p < q,p,qeN} cz U2 and compute the sum S(j>,9)e# 1/2* m t w o different

ways.First, sum horizontally and vertically (Fig. 7)

y oo cc < °° 1

(p,Q)e3Zl k-lj-k+l* 1 ^


Fig. 7.

Next, sum along each line of rational slope and deduce

1ojq

(p/gr'e'duced) " (p/greduced)

The complement of the interiors of the {Iv}peQ/z i

)q 99 _ 10<p/q<l j>\ 0<p/g<l " L

ilq reduced) (p/q reduce

0 = {0>e(R-Q)/Z} U {d±\veQ/Z}.

This set is closed (since its complement is open), and has Lebesgue measure zero.To show that it has Hausdorff dimension zero, we consider {a^Q}j>1.

For each j , o~i is injective on 0 except in a finite set (the end points of certain /„),and Vi,j, (rl(@) 0 <T}(0) is finite. Hence the d-dimensional measure of aj(Q) satisfies

ma(<r'(0)) =

and ...)md(Q).

But U (Tj(Q) cz T, so md(Q) = 0, ~id > 0, and hence 0 has zero Hausdorff dimen-sion. I

Proof of Theorem 4. Let ye(IR — Q)/Z. From Corollary 2 of Proposition 5, we canwrite

where is the sequence of truncations of the continued fraction of v

"*2j a- "~2j+l a+ <Z ft <: ft~

d-

where


(a) If {cij}jî is not bounded, then given any c > 0 it follows from the inequalityabove that there exists a sequence of rationals {Nk/Dk}k>1 such that

and hence 6V is Liouville.(6) (i) If {dj}jg. j is bounded, but infinitely many of the «3- are greater than 1, then

there exists a subsequence with at ^ 2 and hence (for given c > 0) rationals {Nk/Dk}with

a ^k c

and so dv is not 3-Diophantine.(ii) If a} = 1 for j sufficiently large, then qi+1 = q^+q, and

h m ^ = y = -^—-— ~ 0-618.

Hence

i.e.

6--k}

" D<

for given c > 0 and e > 0, and j sufficiently large, and so 6V is not (2 + y — e)-Diophantine. I

Remark. For the golden mean

1v =

1 +1

we have Pi

1 + 1/(1 + ...)

p 2 _ 1 p3 _ 2 p4 _ 3

~~ = 101, 03+ = ToTTo.0r l, ^ 10, 0r

This sequence can be generated recursively, by the substitution rules 1 -> 10, 0-^1.More generally, it is not to hard to prove, using Proposition 5, that d^.a . .a > comesfrom 6t > by the same substitution rules, and so does &},.„. .„ <. froma—( 7 ( a 1 ; . . . ; a 2 n + 1 ) -

Finally we remark that for v the golden mean, the continued fraction expansion of

6»p=(l;2;2;22;23;25;28;21 3;. . . ;2a ' . ; . . .) ,

where {an} is the Fibonacci sequence.

Part 2. Applications and generalizations

1. The square root map: an archetypal example of mode-lockingFor each 6 with 0 < 0 < 1, let Sq r6 denote the map of the complex plane C to itself

defined by sending each 2 to a value of Vz having 0/2 ^ Arg y/z =% 0/2 + §. Note thatSq rg is single-valued and continuous except along the half-line Argz = 0, where it istwo-valued.

470 S H A U N BTJLLETT AND P I E R R E T T E SENTENAC

By an attractor for Sqr5, we mean a minimal closed invariant subset Ae a C, witha neighbourhood TJe such that Sqrfl(C7e) <= Ue and Ag = f]^=0(Sqvg)

n (£/#).

THEOREM 5. For each 6 with 0 ^ 6 < I, the map Sqre has a unique attractor Ae. Themap Sqrfl preserves cyclic order on the attractor and restricted to it has rotation number1 — v where v is the rotation number corresponding to 6 in the devil's staircase {Theorem2, Part 1).

Ifd~ < 6 < &l (in the case where v is rational), the attractor is the set A" (Theorem 1).If 6 = 0±, then Ag = A\ (Theorem 1).

Proof. This is an almost immediate corollary of Theorem 2. Any attractor forSq re must lie on the unit circle (since Sq rg is expanding elsewhere) and must there-fore lie on the semi-circle {z: \z\ = 1,6/2 < Arg2 ^ 6/2+\}. Furthermore, it must beinvariant under z\^z2 (the inverse of Sqr9). Hence it is one of the sets A", A"+, A"_.Since these are also invariant under Sqr^ for the appropriate values of 6, we are done.The rotation number is 1 — vg since Sqre is inverse to ZM>Z2. I

The graph of the function assigning to each 6 the rotation number of thecorresponding attractor for Sqre is the devil's staircase illustrated in Fig. 1.

I t is instructive to consider the above theorem in terms of the symbolic dynamicsof the inverse shift a"1. Each point of the circle is uniquely specified by its argument,te IR/Z. The map ZH> \/Z acts as the inverse shift on the binary expression for t, andthe two values of \/z are represented by the two possibilities, 0 and 1, for the newdigit inserted at the start of the shifted word. Restriction to the (almost) single-valued branch Sq r0 imposes a rule for the choice of new digit, namely ' choose thedigit which gives a new number t' with 6/2 < t' =$ (6+ l)/2 in IR/Z'.

Translated into these terms, the theorem tells us that when we repeatedly applythe rule for Sqre to any initial binary expression, in the limit what is produced is asequence of 0s and Is which depends only on 6. Moreover this limit sequencerepresents an ordered orbit on IR/Z, with a well-defined rotation number 1 — vg. Thusiteration of Sq vg, from any random initial value, is a mechanism for manufacturingthe devil's staircase (Fig. 1). The algorithm given in the proof of Theorem 3 told ushow to compute d± or 6V for a given v: the theorem above gives us a practical wayof computing the inverse, that is vB for given 6.

Examples.(1) If 5 < 0 < §, then all orbits of Sqr^, tend to the period 2 orbit

For instance, with 6 = \ (so the rule is 'choose a new digit to give a value ^ \ and^ f ) , and with initial point x0 = |, we obtain an orbit

xx = 0011,

x2 = 01011,

tending in the limit to {to,t^, the periodic orbit A" for v = \.


(2) If f < 6 < §, then all orbits of Sqr9 tend to the period 3 orbit

h = o l i o ( = §)•

This is the orbit A" for v = §, but the action of Sqrfl has rotation number 1 —§ = §.If 8 = f, we have an alternative choice for the image of tx, namely

but then ^ has subsequent orbit tending to {t0, tx, t2} = A213. In this case, the attractoris A2!3.

Similarly, if 6 = §, we have A2!3 in place of A213 as the attractor, since t2 has analternative image

<i = 0-iTTo ( = ±f)

which again has subsequent orbit tending to A2'3.The mode-locking behaviour of Sqre, that is the tendency to stick at rational

rotation numbers as the parameter 6 is varied, is typical of the generic phenomenonof mode-locking for continuous circle maps ([1], [19]). Although Sqre itself isdiscontinuous, it has a continuous inverse Sqs: U/Z-^-U/Z defined by

otherwise

and this is the map we shall work with.A very similar analysis to that for z>-> y/z can be applied to zt->z1/A (A > 1) if we are

careful about how we interpret z1/A. Define SA g: U./Z-+U./Z by

otherwise.

Then S2 g is just the map Sq9 defined above. However, for a complete analysis ofmode-locking, it is desirable to have a family tending to the one-parameter family ofrigid rotations as A tends to 1, and *SA g has the disadvantage of tending to theidentity for all 6. We therefore modify the definition to

The map T2 g is equal to Rgf2S2tgRg/2, where Rg/2 denotes the rigid rotationth>t+ d/2(mod 1) on IR/Z, so T26 has the same dynamics as Sqg. Moreover, when Atends to 1, the map Txg tends to Re/2 rather than the identity.

Each Tx e is a continuous monotone degree 1 circle map and therefore has a well-defined rotation number /ix g. The following theorem was first proved by Veerman([19], [20]).

THEOREM 6. For each real A > 1, the map 6>->nx g has graph a devil's staircase. Inparticular, for each such A, the rotation number /iK g is rational for a set of 6 of fullmeasure.


e/2 + I i

e/2

l/A

Fig. 8.

Proof. Consider the lift fAg of TA_Ag A_e to the universal covergraph illustrated in Fig. 8.

The rotation number of Tki e is given by

of IR/Z. This lift has

" •

and is clearly a monotone increasing function of 6, for fixed A, since increasing 6moves the whole graph vertically upwards.

If there exist positive integers p and q with

(*)

the orbit of 0 has period q and the rotation number is p/q. Denote by 6p/q the valueof 6 such that the left-hand inequality of (*) becomes equality and 6^lq the value of6 where the right-hand inequality becomes equality. Note that by the monotonicityof {Txe)

q as a function of 6 for each q, the values d±/g exist and are well-defined foreach p/qeQ/Z. For 6~/g < 6 < 0+/9, increasing 8 by 8 moves (^>e)

9(0) a distance

It follows that ffr ft- _1\_2(A-1)2

A; A(A«-I)"

We compute the sum of the lengths 8p/g — 6p/g over all ̂ /g in reduced form, with0 < p/q ^ 1, in the same way that we computed the corresponding sum in the caseA = 2 (Theorem 2). As there, let 0t= {{p,q)\p <q,p,qeN}. Then

but

(A- l ) 2

(A- l ) 2

A«

V VZJ ZJ

s0<p/g<l

(p/qreduced)

(A-l)2

V ( A -m A"

Vfc-i

I)2

A -A*

(P/<?

1 A - l

A - l

y (A-

reduced)

I)2

"I


o i 5 \ o

Fig. 9. 'Arnol'd tongues' for the maps 7̂ e.

= 0.1000

v0 = 0.0001

Fig. 10.

474 SHAUN BULLETT AND PIERRETTE SENTEKAC

„ „ 2(A-1)2 2Hence £ y^—— = y

(p/qreduced)

and so X,o<p/«sa ( 1) A A

(p/g reduced)

Thus we know that for each fixed A, the rotation number /ix e is rational except ona set of values of 6 of measure 0. It remains only to show that all irrational valuesare achieved on this set. For a given ve(M — <Q)/Z, consider {8+/g: p/q < v) and{0~/g: p/q > v). The least upper bound of the first set and the greatest lower boundof the second coincide, since rational rotation numbers occupy full measure. Denotethe common limit by 0v. By monotonicity of /tA_fl as a function of 6, we have

We remark that as A -> 1, the map TA e tends to the rigid rotation Ee/2 in the C°-topology. For each A > 1 the theorem gives us mode-locking on a set of values of 6 offull measure, unlike the C00 case [1] where mode-locking occurs on a set of decreasingmeasure as one approaches rigid rotations. Note also that as q increases the value ofA for which 8p/g — 6~/g has maximum length becomes closer and closer to 1 (Fig. 9).

2. Orbits with sequences of rotation numbers: 'Douady tuning'

The smallest orbit of a: t>-> 2£(mod 1) on IR/Z which is not ordered is the period 4orbit

to = o-ooii = ±,

t, = o-oTIo = f,t2 = 01100 = | ,t3 = o-iool = f.

However, under a2: <H*4£(mod 1) this splits into two ordered orbits {to,t2} and{tlt t3}. Moreover one of these, {tx, t3}, lies on a quarter circle.

Any orbit of the ft-shift <rn on IR/Z which remains within an interval of length 2~"has its cyclic order preserved by an, since crn multiplies lengths by 2". But theconverse is false: in the example above {t0, t2} has its order preserved by cr2 but it doesnot lie within a quarter circle. Thus the situation is not as straightforward as for thesimple shift a.

We may think of {to,t1,t2,t3} above as a 'satellite' of the period 2 orbit {0-01,0-10}of cr. The following is an example of a period 12 orbit which is a satellite of a period4 orbit u0 = 0-000100010010 = ^ .

This has orbit under a illustrated in Fig. 10. The orbit {v0, vlt v2, v3} in the Figure hasperiod 4.

Note that the points {vo,vltv2,v3} partition the a orbit of ti0 into 4 sets:Ao = {uo,u4,us}, Ax = {uûÛg}, A2 = {u2>u6,u10}, A3 = {u3,u7,un}. Each of thesesets is invariant under cr4 and cr maps each A{ bijectively to Ai+1 (where i +1 is takenmod 4).

More generally, for v = p/q, the orbit of 6~ partitions T into q intervals

ô = [&;, Of], h = <rVo, -JQ-i = ^ { h )


Fig. 11.

of lengths l / ( 2« - l ) , 2 / (2 9 - l ) , ...,2«"V(29-1) respectively which have a rotationnumber c i n a certain sense.

Consider the restriction of crQ to Io and let Jo denote the subset of/0 invariant undercr9. Then Jo is a Cantor set obtained by removing intervals as illustrated in Fig. 11.

Here d+ and d~ are the blocks of q digits which define 6^/g = d+ and 8~/g = d~ (seeProposition 5, Part 1).

Let Jo denote the space obtained from Jo by identifying end points of each gap.Then, if we define h to be the map JQ-S-T sending each point to the binary number(mod 1) represented by its coding sequence, and we let Jo denote classes of points withthe same image, the following lemma is self-evident:

LEMMA 1. h induces a bijection Jo -on! I.

• T conjugating the action of cr9 on Jo to that of cr

This allows us to speak of the rotation number of an orbit of cr9 on Io and to makethe following inductive definition:

A closed subse t 4 c R has a sequence of rotation numbers (vlt v2, •••,^n) f ° r °"> where

- if 71 = 1, and A has rotation number j ^ for a, or- if n > 1, vx = pjq^ i c ^ U / j U ... U -4,-i' °" m a P s e a c n ^-i = ^ n A bijectivelyto Ai+X and cr*1 acts on Ao with sequence of rotation numbers (v2,..., vn).

In particular, the example in Fig. 10 has sequence of rotation numbers equal to

THEOREM 7. For each (v1,..., vn) with iêQ/Z (1exists a unique minimal closed cr-invariant set A

i ^ n—l) and uneU/J., thereR/Z with rotation sequence

Proof. Consider the following explicit algorithm for the construction of A, when vn

is rational, as the orbit of a point 6jv „ ̂ obtained by successive modifications of 6~ ,or, when vn is irrational, as the closure of the orbit of a point djv v , obtained fromvn in the same way. Let d~ _ denote the binary block of length qn_x which gives &lwhen repeated, and let d* _ denote the corresponding block for 0+ . The first stepis to replace all occurrences of 0 in d~ by copies of d~ __ and all occurrences of 1 bycopies of d* _. We now repeat the process, replacing all 0s in the new word by d~ _and all Is by d* _ . After n—l steps, we obtain a binary expression for a point 6 whoseorbit is periodic of period q1q2---qn if vneQ/I., or non-periodic if vne(M — Q)/Z. Ineither case, A is the closure of this orbit.

PSP115

476

0010

SHAUN BTJLLETT AND PIERRETTE SENTENAC

ooolo

0100

00001

Fig. 12. cr2\A0 has sequence of rotation numbers (J,|).

We show by induction on n that the point 6 constructed above has sequence ofrotation numbers (y1,...,vn), using the following elementary result (the proof ofwhich is left as an exercise for the reader).

L E M M A 2. Let $>p/g denote the operation of replacing all 0s in a word by copies ofd~/q

(the periodic block for dp/q) and all Is by copies of dp/g. Then, for any x (i) the orbit ofQ>p/q{x) under a9 lies between 6p/g and 8p/g, and (ii) o-g(<I>p/g(x)) = (t>p/g(crx) I.

As an inductive hypothesis, we suppose that the point constructed by applying thefirst n — 2 modifications to 6V has sequence of rotation numbers (v2,..., vn). We mustshow that applying the final modification Op /g yields a point 6 having sequenceof rotation numbers (i^, ...,vn). But Lemma 2(i) guarantees that the closure A ofthe orbit of 6 has the property that <x maps each Ai bijectively to Ai+l (i+1 takenmodq^ and Lemma 2(ii) guarantees that a9 has sequence of rotation numbers(v2,..., vn) on Jo. Note that ^p/g is an inverse to the composite mapLemma 1.

The proof of uniqueness of A is analogous to that in Theorem 1. I

umJ0-»T of

Example 1. For the sequence of rotation numbers (\, \, \), we start with 6\/z = 0001and we replace all 0s by dj~/4 = 0001 and all Is by copies of df/4 = 0010 and again all0s by dy2 = 01 and Is by d+/2 = 10 (Fig. 12). This yields:

= 0000100010010,

= 0010101100101011001011001.

Ordered orbits of the shift

Fig. 13.

Example 2. The Morse number 00110100110010110 ... is the limit of %>,>,...,.

Remark 1. For vn rational, 0^ _ r , is defined by applying exactly the samesequence of modifications to 6* . In C— M (M the Mandelbrot set), the external raysof argument #(*_..._„ >, vn rational, ov Q(v v y vn irrational, land on dM at a pointaccessible from the main cardioid Mo by a finite number of crossings of boundaries ofadjacent hyperbolic components of the interior oiM. The sequence (vt,..., vn) tell usthe sequence of internal rays to follow. Now for any hyperbolic component IF of theinterior of M there is a smaller copy Mw oiM, within M, in which W corresponds toMo, the main cardioid [7, 9]. 'Douady tuning' [7] expresses external angles on dMw

by taking the corresponding angle on dM and replacing 0s and Is by the periodicblocks d~,d+ of the pair of external angles of the 'root' of W. The algorithm aboveamounts to the composition of the sequence of Douady tunings corresponding to theboundary crossings encountered on the path from Mo to W. Beyond the inner regionof M accessible by such finite paths lies an outer region containing small copies of Mconnected to one another and the main cardioid by ' filaments'. Douady tuning seemsthe appropriate language in which to describe 'renormalization structure' in thisouter region too, but we do not know of any 'rotation number' interpretation of thesequences arising there (that is to say all sequences of 0s and Is other than those wehave been considering). See [3, 4, 5, 16] for more details of the combinatorics of M,and [13, 14] for the combinatorics of polynomials of degree higher than two.

Remark 2. The definition of 'sequence of rotation numbers' we have given is notpurely combinatorial (it involves specific points 6^/g and not just the order of pointsof the set A c T). However, if we relax the definition, to require only that A can bepartitioned into sets Ao, ...,Aq _lt each invariant under cr91 with sequence of rotationnumbers (v2, ...,vn), and permuted among themselves by a with rotation numberP1/q1, then there may be several candidates for the overall order around the circle.For example, the two orbits illustrated in Fig. 13 are both candidates for (|, 5).

However, we shall see in the next section that in this example only one of theseorders (that on the left) is realizable by an orbit of a. I t is possible that the abovecombinatorial definition of a sequence of rotation numbers is sufficient for Theorem7 and that in general only one of the possible orders for a given (vlt..., vn) is realizableby an orbit of a. We leave this as an open question for the reader.

478 SHAUN BTJLLETT AND PIERRETTE SENTENAC

x4'

Fig. 14.

3. Orders of points around T realizable as orbits of a

Certain orders of points around T cannot occur for orbits of cr. For example, theorder indicated in Fig. 14 is impossible, as one can easily check by listing all period5 binary sequences.

Let X = (xo,x1,x2,...), Y = (yo,yvy2,...) be finite or countably infinite sequencesof distinct points of T, both of the same cardinality. We say that X is equivalent toY if, for all triples (i, j , k) of positive integers, (xt, x}, xk) and (yt, yp yk) define the sameorientation on T, and we say that X is realizable if it is equivalent to an orbit of a.Note that we say nothing about the positions of points of accumulation of X. Forexample if X is the orbit of 0 under t M> t + v with v e R — Q (an irrational rotation ofT), then X is dense in T, but its realization as an orbit of a is certainly not, since thatorbit is contained in a semicircle. Thus the bijection between an infinite X and itsrealization need not extend to a homeomorphism of T.

Let s denote the successor function xi t-+ xi+1 (where i + 1 is taken mod |X| if X isfinite). The key to realizability is the following condition: 3zeT— X such that eachof the two restrictions

s : X n ] z , x o + l ] îs order-preserving.

This should be compared with the definition in Part 1, Section 1, of an ordered set,where we required s to preserve order on the whole of X.

Condition (S) is clearly necessary for realizability, since if X is an orbit of a we maysimply take z = x0 + 1/2 and (S) will be satisfied. As we shall see, it is also a sufficientcondition for realizability in the case that X is finite, and with a minor modificationit also becomes sufficient when X is infinite. Before formally stating these results(Theorem 8, below) we construct a candidate binary sequence for the realization.

Let X satisfy condition (S), and let X denote the set of all points of U which projectto X under n: U -> T = IR/Z. Let xt denote xi regarded as a point of [x0, x0 +1[ c 05.Define h: X (] [xo,xo+l]-+X f| \xl,x1-\-2] by

xt i-> first lift of xi+l in [xlt xl + 2] if xt e [x0, z[,

xt^~second lift of xi+1 in [x1,x1 + 2] if xte]z,x0+ 1],


Extend h to a monotonic map [xo,£o+ l]-> [x1,x1 + 2] by the same method as weused in the proof of Lemma 1 (Part 1: Section 1), that is to say (essentially) linearinterpolation, and further extend to a monotonic map A:IR->IR by periodicity(sending [xo + n, xo + n+l] to [x1 + 2n,x1 + 2n + 2]). If Xis finite h will be continuous,while if X is infinite h may have jumps. However, in either case h is monotonic, andis the lift of a degree 2 map h of T.

If h is continuous there exists at least one point a. e U with h(a) = a, and if h is notcontinuous we can still ensure that there exists such an a by (if necessary) inserting anopen interval into T— X at the point where the graph of h jumps the diagonal, andlinearly interpolating along this interval: this alters the topology of X but not itscyclic order. Similarly we can find a point ft with h(f$) = 6L+ 1, again after alteringthe topology of X if necessary. Note that: (i) a and /? project to points a,/?eT withh(a) = h(P) = a; (ii) h restricts to order-preserving maps [<*,/?]->[a,a+ 1] and[/?, a + l ] ^ [ a , a + l ] . Thus a and /? behave very much in the same way for h as do 0and 1/2 for a. To the point x0 of X we now associate the sequence

where ^(a;0) = 0 if x3e[a,/?[ and t}(x0) = 1 if aê[/?, a + l [ . We remark that thesequence T(X0) depends only on the choice of the partition point zeJ — X (if there isa choice) and not on the particular map extension h, since the positions of a and /?in the cyclic order on X depend only on z.

THEOREM 8. (i) (a) If X is finite then X is realizable if and only if it satisfiescondition (S); (6) if X is infinite, X is realizable if and only if it satisfies condition (S),with the corresponding sequence T(X0) not preperiodic.

(ii) In both cases, the binary sequence T(X0) represents a point of T which has orbitunder a realizing X.

Proof. As we have already observed, if X is an orbit of a, we may satisfy condition(S) by taking z = xo+ 1/2. The points a and [5 constructed as above will then be thepoints 0 and 1/2 on T, and the sequence T(X0) will be the binary expression for x0. IfX is infinite, this sequence cannot be preperiodic. Hence the conditions in (i) arenecessary.

For the converse, if X satisfies (S) we may associate to each xt eX a binary sequenceT(xt) which codes its [a, /?[, [/?, a + 1[ itinerary, in just the same way as we coded theitinerary of x0. Since a acts as the shift on T(XJ), if we are to prove that {T(xt)}i>0 isa realization of X it will suffice to show that

(where ' < ' is defined by cutting open the two circles at a and 0 respectively).If x( 6 [a, /?[ and xt e [/?, a +1 [, then by comparing the first digit of r{xi) with that of

T(X}) we have that T(xt) < T(XJ). But if both xl and x} lie in [a, /?[ (or both in [fi, a+ 1[)we know that the order of (a, xt, x}) (or (xu xp a +1)) is preserved by h, since h mapseach of [a,/?[ and [/?, a + l [ monotonically onto the whole of T. I t follows that tocomplete the proof it suffices to show that if xi 4= x} their associated binary sequencesT(xi),T(xj) differ at some digit. In the case that X is infinite, this is implied by thehypothesis that T(X0) is not preperiodic. In the case that X is finite it follows withoutfurther hypothesis by the following argument. If two points of X have the same

480 SHATJN BULLETT AND PIERRETTE SENTENAC

xi+2

xo + 2

x5 + 1

x4 + 1

X2+l

x4

x\

'7/

7 1

7

A i | a

r

V

//

/

x0 x2

Fig. 15.

x\ x4 X5 X6

itinerary, then so do all points of X between them, and there is therefore an adjacentpair of points of X with this property. Such a pair is mapped to itself by hn (wheren is the number of points in X) and indeed the line segment joining the pair is mappedto itself by hn. Since h is transitive on X this implies that h has degree 1, contradictingthe fact that we constructed it to have degree 2. I

Example. Is X pictured in Fig. 15 realizable ? The answer is yes. We construct hand calculate T(X0) = 0011000 = 8/21.

Remarks. 1. We need the condition of non-preperiodicity of the itinerary, in thecase of infinite X, to avoid examples such as {xn = 2~n}n>0, which is not realizable asan orbit of cr.

2. The realization of a finite X satisfying condition (S), as an orbit of a, can alsobe achieved by 'renormalization'. For n = \X\ the points hn(x0),^"(^j),...,hn(x0+ 1),in (R, are spaced along an interval of length 2" in the same order as xo,£l,...,xo+i,and they are mapped to one another in the same way by h. However, rescaling this


long interval by dividing by 2n replaces h by a map very close to a linear map (ofslope 2), and repeating the process yields (in the limit) a set of points in the samecyclic order as X and mapped to one another by a.

We are deeply indebted to Adrien Douady for many helpful discussions. The firstauthor would also like to thank Franco Vivaldi and Ian Percival for comments andsuggestions on a very early version of some of this work.

REFERENCES[1] V. I. ARNOL'D. Small denominators. I. Mapping the circle onto itself. Akad. Nauk. SSSR, Ser.

Math. 25 (1961), 21-86, and Geometrical methods in the theory of ordinary differentialequations (Springer-Verlag, 1983).

[2] P. ATELA. Bifurcations of dynamic rays in complex polynomials of degree two. Erg. Th. Dyn.Syst. 12 (1992), 401-423.

[3] C. BANDT and K. KELLER. Symbolic dynamics for angle-doubling on the circle. I. Thetopology of locally connected Julia sets. In Lecture Notes in Mathematics 1514 (SpringerVerlag, 1992), 1-23.

[4] C. BANDT and K. KELLER. Symbolic dynamics for angle-doubling on the circle. II. Symbolicdynamics of the abstract Mandelbrot set, preprint.

[5] C. BANDT and K. KELLER. Symbolic dynamics for angle-doubling on the circle. III. Sturmiansequences and the quadratic map, preprint.

[6] B. BRANNER. The Mandelbrot set. In Proceedings of Symposia in Applied Mathematics 39(1989) (AMS Providence, Rhode Island).

[7] A. DOUADY. Algorithm for computing angles in the Mandelbrot set. In Chaotic Dynamics andFractals (Academic Press, 1986).

[8] A. DOUADY and J. H. HUBBARD. Iteration des polynomes quadratiques complexes. C.R.Acad. Sci. Paris, t294, Seri. I (1982), 123-126.

[9] A. DOUADY and J. H. HUBBARD. Etude dynamique des polynomes complexes. (Publ. Math.Orsay I 1984, II, 1985).

[10] A. DOUADY and J. H. HUBBARD. On the dynamics of polynomial-like mappings. Ann. Sci. Ec.Norm. Sup. (Paris) 18 (1985), 287-343.

[11] J. M. GAMBAUDO, O. LANFORD and C. TRESSER. Dynamique symbolique des rotations. C.R.Acad. Sci. Paris, t299 (1984), 823-825.

[12] P. A. GLENDINNING and C. T. SPARROW. Prime and renormalisable kneading invariants andthe dynamics of expanding Lorenz maps. Physic 62D (1993), 22-50.

[13] L. R. GOLDBERG. Fixed points of polynomial maps I. AnnEc. Norm. Sup. 25 (1992), 679-685.[14] L. R. GOLDBERG and J. MILNOR. Fixed points of polynomial maps II. Ann. Ec. Norm. Sup. 26

(1993), 51-98.[15] J. H. HUBBARD and C. T. SPARROW. The classification of topologically expansive Lorenz

maps. Comm. Pure App. Math. 43 (1990), 431-444.[16] P. LAVAURS. Une description combinatoire de ['involution definie par M sur les rationnels a

denominateur impair. C.R. Acad. Sci. Paris, t303 (1986), 143-146.[17] M. MORSE and G. A. HEDLUND. Symbolic Dynamics II. Sturmian Trajectories, Am. J .Math.

62 (1940), 1-42.[18] K. F. ROTH. Rational approximations to algebraic numbers. Mathematika 2 (1955), 1-20;

corrigendum, ibid 1 (1955), 168.[19] J. J. P. VEERMAN. Symbolic dynamics and rotation numbers. Physica 134A (1986), 543-576.[20] J. J. P. VEERMAN. Symbolic dynamics of order-preserving orbits. Physica 29D (1987),

191-201.

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