STURMIAN MINIMAL SYSTEMS ASSOCIATED WITH THE ITERATES OF CERTAIN FUNCTIUNS ON AN INTERVAL John C. Kieffer* Dept. of Electrical Engineering University of Minnesota Minneapolis, MN 55455 I. Background. (The reader who is not interested in this section may skip directly to the next section.) Consider the communication system given by the following block diagram. message X source... > i nco e I, T (Yo,Y1... )........... > ---> (Xo, XI... ) The message source generates an analog message which we take to be a continuously distributed random variable X with finite variance. Using a given encoding scheme, a digital representation of X is produced consisting of a random sequence (Yo,YI... ) of symbols from a binary set of reals la,b~. This encoder output sequence is then noiselessly transmitted to some user, who decodes this sequence ^ ^ into a sequence (Xo,X I... ) of reproductions of X by means of a linear filter h. Mathematically, we may think of the filter h as an absolutely summable sequence of real numbers h = (hu,h I... ), which operates on the encoder output sequence via discrete-time convolution: i )~i = S hn_iy n, i=o,l,... n=o The fidelity with which the filter h generates successive approximations {X~i} to X is measured by the number I n-i a h = lim sup ~ z E[Xi-X] 2. n ~ i=o One filter h I will be regarded as being better than another filter h 2 if ~hl < Ah 2. For the given encoding scheme, one would like to select a filter h* for which a h is at or near its minimum value over all filters h. For this purpose it would be useful to solve the problem (originally considered by R. M. Gray [2] for a special type of encoder) of obtaining an explicit formula for a h as a function of the filter h. *Research supported by NSF Grants ECS-8300973 and ECS-8501068, and by the Joint Services Electronics Program under Contract N00014-79-0424.
355 Of particular interest in communication engineering is the case in which the encoding scheme is of "feedback type". By this we mean that there is a pair L,U of measurable functions from (--, )X[-1,1]~[-1,1] and a random sequence of "feedback states" So,S I... from [-1,1]~[-1,1] such that S O is constant a.s.; Si+ I = L(X,Si), S i < O; Si+ I = U(X,Si), S i ) O; Yi = a, S i < O; Yi = b, S i ) O. Gray [2] developed a method for finding ~n which works for the feedback type encoding scheme known as sigma-delta modulation. The author [4] gave a probability method for finding a h which is valid for any encoding scheme, for which, for almost every value x of X, the sequence of encoder outputs YO = YO, YI = YI,"" produced when X = x is a typical sequence for a stationary process whose spectral measure can be found. This requirement can be shown to be satisfied if the iterates of the following function f defined on [-i,i] are sufficiently well-behaved: f(.) : L(x,,) on [-i,0); f(.) : U(x,.) on [0,1]. The main resu}t of this paper, Theorem i, gives the desired behavior of these iterates so that the probability method in [4] can be carried out for a wide class of encoding schemes with binary output of feedback type. The interested reader is referred to [4] for the details of this application. 11. Statement of Main Result. Fix for the rest of this paper a function f: [-1,1]~[-1,1] such that: 2.1 Jf(xl)-f(x2) I < JXl-X2J, -I < x I < x 2 < O; 2.2 If(xl)-f(x2) I < JXl-X2j, 0 < x I < x 2 < 1; 2.3 For any x 0c[-1,1], there exists 6 > 0 such that f(x) ~ f(xo), if x~[-1,1] and 0 < jx-xoj < 6. Conditions (2.1)-(2.2) state that f is a nonexpansive function when restricted to [-1,0) and also when restricted to (0,1]. Condition (2.3) is satisfied by functions which are piecewise strictly monotone or piecewise analytic. Let If = {xm[-1,1]: fi(x) ~ O, i=0,i... }, where fo denotes the identity function and for each positive integer i, fi denotes the i th iterate of f. It is easy to see that If contains all but countably many points of [-1,1]. For xmlf, the binary seqeunce (YO,Yl... ) such that Yi = 0 whenever fi(x) < 0 and Yi = i whenever fi(x) > O, is called the itinerary of x under f [I]. We want to examine the dynamical behavior of the itineraries of the points in If. To this end, if x~if, let S(x) be the symbolic system generated by the itinerary of x under f, that is,
356 the set of all binary bisequences whose finite blocks of consecutive symbols are blocks appearing in the itinerary. Our main result is the following. Theorem 1. For each x~if, S(x) is a Sturmian minimal system. We shall eventually prove a result (Theorem 1', Section IV) slightly stronger than this. Ill. Preliminaries to Main Result. sequences (UI,U2,...), of length at least one, such that Let~be the set of all finite or infinite 3.1 Each U i is a set consisting of two binary words of finite length, one beginning with "0" and the other with "i". 32 ul : {0,I}. 3.3 If Ui,Ui+ 1 are consecutive terms from the sequence, and 3.4 U i = {wo,wl}, then Ui+ I = {wowl,wl} or {WO,WlWO}. If the sequence is infinite, then the minimal length of the words in U i tends to infinity with i. We now define some symbolic systems based on the sequences in~. If U E~is infinite, let S*(U) denote the symbolic system consisting of all bisequences which, for each i ) 1, can be partitioned into words from the i th set in U. If U E~is finite, with final set {WO,Wl}, where w 0 begins with "0" and w I with "1", let Si(U), i=0,i, be the symbolic system consisting of w i and its shifts, where, if w is a binary word of length n, w denotes the periodic bisequence (v i) such that (Vnj,Vnj+l... Vnj+n+l) = w for each integer j. of form S*(U), So(U), SI(U), U~ Let X~be the family of all systems The proposition at the end of this section states that every system in'is a Sturmian minimal system [3][5]. To prove this, we shall need the following lemma, whose simple proof is omitted. (In what follows, if i is a symbol from {0,1 1, let Tdenote the other symbol from this set. Furthermore, if S is a symbolic system, the S-words are the finite words that appear in the bisequences in S.) Lemma. Let B be the set of all finite binary words, regarded as a semigroup under concatenation of words. For i = 0,1, let B i be the subsemigroup of B consisting of all words in B which end in i and do not contain the block ~"~, and let ~Pi: Bi+B be the homomorphism such that @i(i) = i, ~(Ti) : T. Let (UI,U 2... )~ have length at least two. Let i~{0,i} be the symbol such that every word in Un(n)2) ends in i. Then ( i(u2), i(u3)... )~. Finally, if S1,S 2 are the symbolic systems S I = S*(U 1... ), S 2 = S*(@i(U2)... ) every Sl-WOrd ending in i maps under i into a S2-word.
357 0 Proposition. Every system in~x~is a Sturmian minima] system. Proof. We omit the easy proof that every system in'is minima]. We now use mathematical induction to prove the property that for n = 1,2,... every S c~is such that any two S-words of length n have weight (i.e., number of ones) within one of each other. (This will mean that every S c~is Sturmian.) The property is certainly true for n = 1. Fix N > 1 and adopt as induction hypothesis the statement that if 1 < k ~ N-1 and S~ then any two S-words of length k have weight within one of each other. We need to prove that this holds for k = N as well. Suppose this is not true for k = N. Then we may choose S~and two S-words x,y of length N which do not have weights within one of each other. We must have S = S*(U), where U~has length 2. Consequently, either O0 is not an S-word or 11 is not. Without loss of generality, we may suppose that the former is true. Then x,y can be assumed to take the form x = OuO... y = lui... where u is some block of length at least one which must begin and end with one. (Thus N ) 3.) If u = i, then 0101 and 11 are both S-words, whence, by the Lemma, ~1(0101) = O0 and @1(11) = 11 are S'-words for some S'~ violating the induction hypothesis. Thus, u = lv, where v has length at least one and ends in one. We again arrive at a contradiction, since then 01v01 and lvi are both S-words, whence their ~1-images O@l(V)O and 1~1(v)i would have to have weights within one of each other, as their common length is less than N. 0 Remark. It is true that~c~coincides with the family of Sturmian minimal systems. We do not need this stronger result for the purposes of this paper. IV. Proof of Main Result. Let us agree to call two Sturmian systems S1,S 2 complementary if there is a finite sequence U such that So(U) is one of S1,S 2 and SI(U) is the other. Our main result is: Theorem I'. One of the following holds: (i) There is a Sturmian minimal system S such that S(x) = S for every x~if; or (ii) There are complementary Sturmian minimal systems S1,S 2 such that {S(x): x~if} : {$I,$2}. Furthermore, (ii) can't occur, i_ff, in addition to what was previousl ~ assumed about the function f, one has: (iii) f is strictly increasing on [-1,0) and on (0,1]; (iv) If -I < x < O, then fn(x) ) 0 for some n ) 1,
358 and if 0 < x ~ i, then fn(x) ~ 0 for some n ) i. holds. Proof. The first part of the proof is devoted to showing that (i) or (ii) Let~be the family of all quadruples (I,J,m,n) such that 4.1 I,J are subintervals of [-1,0), (0,1], respectively, each having zero as an endpoint; 4.2 m is a positive integer such that O~Efk(1), 0 ~ k (m-l, and fm(1)ciu{o)uj; n is a positive integer such that @~fk(j), 0 ~ k < n-l, and fn(j)clu{o}uj. Let'~/~ ' be the set of all (l,j,m,n)~'~/)such that 4.3 fm(1) has nonempty intersection with J and fn(j) has nonempty intersection with I. 4.4 Either fm(1) or fn(j) has nonempty intersection with both I and J. Let {li,ji,mi,ni}~:o be the sequence from~/~such that 4.5 (Io,Jo,mo,no) = ([-1,0),(0,1],I,1); 4.6 If (li,ji,mi,ni)e~' and fni(ji)c.l i, then li+ 1 = I i, Ji+1 = Ji, mi+1 = mi, ni+1 = ni + mi; 4.7 If (li,ji,mi,ni)~ and fmi(li)cj i, then Ii+ 1 = I i, Ji+1 = Ji, mi+1 = mi + ni, ni+l = ni; 4.8 If (li,ji,mi,ni)~' and oefmi(li)~fni(ji ), then li+ I = [fmi(li)(jfni(ji)]~'~[-1,0), Ji+l = [fmi(li)~fni(ji)]~(o,l], 4.9 If (li,ji,mi,ni)#~', then li+ I = I i, Ji+1 = Ji, mi+1 = mi, ni+l = hi; mi+ I = mi, ni+1 = n i. Note that every xclf~i i has an itinerary which starts off with the same word of length m i, which we will denote by wo(mi); similarly, every x~if~j i has an itinerary which starts off with a word of length n i that we shall denote by wl(ni). For each i, let U i = {wo(mi), w1(ni) }. Since every x~if satisfies fn(x)~liuji for some n ) O, every point in If has an itinerary which, beyond some point, can be partitioned into words from U i. Also, the words in the set Ui+ 1 are formed from the words in the set U i in one of the following three ways: wo(mi+1) = wo(mi), and wl(ni+l) = w(ni) ; or wo(mi+1) = wo(mi)wl(ni), and w1(ni+l) = wl(ni); wo(mi+l) = wo(mi), and wl(ni+l) = w1(ni)wo(mi). Thus, if the lengths of wo(mi),w1(n i) both go to infinity with i, IS(x): x~ifl = S*(U'), where U' is the sequence in~-obtained by deleting all repetitions from the sequence (Uo,U 1... ). We now handle the case that arises when the length of wo(m i) does not go to infinity with i, or the length of wl(n i) does not go to infinity with i. Without loss of generality, we suppose that the latter is true. If a term (li,ji,mi,ni) from our sequence does not lie in~', we are done. (Either (4.3) or or
359 (4.4) is violated -- if (4.3), then every S(x) is either the system consisting of wo(mi) and its shifts or the complementary system consisting of wl(ni) and its shifts; if (4.3) holds and (4.4) is violated, every S(x) coincides with the Sturmian system consisting of [wo(mi)wl(ni)] ~ and its shifts.) assume that every (li,ji,mi,ni) lies ins. wl(ni)'s, and fix a positive integer i' such that n i : j, i ) i' Consequently, we may also Let j be the maximal length of all the We may assume that the set E is infinite, where E = {i ) i': o cfmi(li)~fni(ji)}. (Otherwise, choosing i* to be a positive integer greater than every element of E we must have fmi(li)cji, i ~ i*, whence, by condition (4.7), every Six) is the Sturmian system consisting of wl(ni,)" and its shifts.) Let ~ be the smallest positive zero of fj. Then ~is in every interval Ji' Suppose fni(ji)~(0,1] = Ji+1, for infinitely many ice. Since fj is nonexpansive on Ji for sufficiently large i, [1] would yield IJi+11 ~ max(~,ijil - ~), for infinitely many i, where l'i denotes length. It would then follow that for some i~e satisfying [1] Ji = (0,~] and fni(ji) = [0,~], whence, for this i, we would have that every S(x) is the system consisting of wl(ni ) and its shifts, or the complementary system consisting of wo(m i) and its shifts. Therefore, we may assume that [i] does not hold. Thus IIil does not tend to zero with i, as otherwise [i] would indeed hold. lengths of the li's. Then for sufficiently large ice, [I] Let ~ > 0 be the limit of the fmi(li)~)ji+ 1 and fni(ji)~li+ I. [2] From this we conclude that ~ ~ B, and that IIi+zl < max(:,ijil - ~) for sufficiently large ice, whence for sufficiently large ice. Thus ~= B and I i = [-~,0), for large i. As, from [2], IJi+ll < llil for sufficiently large ice, we must have Ji = [0,~], for large i. For some i for which [2] holds we may then replace Ji+l,li+l in [2] by Ji,li, respectively, from which we can conclude that every S(x) is the Sturmian system consisting of [wo(mi)wl(ni)] and its shifts. For the last part of the proof, assume now that the function f satisfies (iii) and (iv). We will show that (ii) can't occur. If (ii) can occur, we see from (iv) and the examination of all possible cases in the preceding argument in which possibility (ii) arises, that there must be (l,j,m,n)~such that There exists xclf~if such that fmk(x)cl(k ) 0); there exists yclt'~if such that fmk(y) cl(k ) O, k even),
360 and fmk(y}~j(k ~ O, k odd), or else there exists x,y~j with analogous properties. The preceding statement clearly cannot hold, as fm is nonexpansive and strictly increasing on I. Acknowledgement. The author is indebted to Professors Karl Petersen and Ethan Coven for helpful comments. References 1. P. Collet and J.-P. Eckmann, Iterated Maps on the Inverval as Dynamical Systems, Birkhauser, Boston, 1980. 2. R. M. Gray, "Spectral Analysis of Sigma-Delta Modulation Noise", manuscript. 3. G. A. Hedlund, "Sturmian Minimal Sets", Amer. J. Math., Vol. 66, pp. 605-620, 1944. 4. J. C. Kieffer, "A Probability Method of Solution for a Linear Filtering Problem of R. M. Gray", manuscript. 5. M. Morse and G. A. Hedlund, "Symbolic Dynamics II. Sturmian Trajectories", Amer. J. Math., Vol. 62, pp. 1-42, 1940.