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Interval Exchange Transformations INTERVAL EXCHANGE TRANSFORMATIONS By WILLIAM A. VEECH* 00. Introduction Denote by A,. C_ R m the cone of positive vectors A =(A1,...,Am), A~>0, alli. Let I A I = E Ai, and partition [0, I A I) into m subintervals [/3i_1,/3~), 1 =< i -< m, where /3, =/3,(A) is = ~ 0 i=0, /3, (,x) t AI+-'-+A~ l<=i<--m. ~,, is the permutation group on {1, 2,..., m}. Given A E Am, ~r ~ ~,, set X "/I" = (A 'rg 11, A .fi---12, " * * , a ,rf--lrtl)~ Finally define T = T(~.~), the (A, 7r) interval exchange by the formula (0.1) T.,~)x = x -/3,_,(x) +/3~,_,(;t ~) (x ~ [/3,_1(x),/3, (x))). For typographical reasons we shall, in places where no ambiguity might arise, speak of (A, 7r) or T instead of T(~.,~). The interval exchanges are precisely the piecewise order preserving isometries (with finitely many discontinuities) of intervals [0, ~'), ~- > 0 (z = I A I). The inverse of (A, 7r) is (A ~, ~'-~) (see [8]). Formally, T in (0.1) has m - 1 discontinuities, but there may actually be fewer depending upon zr. In any case, T is right continuous at all points. 9 Research supported by NSF-MCS75-O5577 and an Institute for Advanced Study grant. 222 JOURNAL D'ANALYSE MATHI~MATIQUE, Vol. 33 (1978) INTERVAL EXCHANGE TRANSFORMATIONS 223 M. Keane [8] made the first basic study of interval exchanges, and it will be convenient to recall here certain of his results. Call T minimal if for each x E [0, I A I) the orbit r = {T"x t n ~ Z} is dense in [0, I A I). T satisfies the infinite distinct orbit condition (i.d.o.c.) if (a) r (A)) is an infinite set, 1 _-< i _--<m - 1, and (b) C(/3, (;~)) n r (X)) = ;~, 1 =< i ~ j _-__m - 1. 0.2. Theorem (Keane [8]). If (A, or) satisfies the i.d.o.c., then (A, zr) is minimal. An obvious necessary condition for (A, ~') to be minimal is that 7r not map dny segment {1, 2,... ,j}, j < m, into itself (for then [0,/3j(A)) would be an invariant set). Such a ~" is irreducible. If 7r is irreducible, Keane observes that a sufficient condition for the i.d.o.c, to be satisfied is that A be irrational; that is, that I AI, A1,'-',Am-1 be linearly independent over Q, or, equivalently that all rational relations between A~/IAI,..-,Am/IA I be consequences of EA,/IAI= 1. In particular, 0.3. Corollary. If Ir ~ ~, is irreducible, then for almost all A E Am the (A, ~) interval exchange is minimal. To each interval exchange (A, 7r) we associate the cone E (A, ~-) of (nonnegative) nonzero invariant measures for (A, ~r). E(A, ~r) contains Lebesgue measure on [0, IA I) (and all its positive scalar multiples). We say (A, zr) is uniquely ergodic if E(Jt, ~r) is one-dimensional; that is, if every invariant measure for (A, 7r) is a multiple of Lebesgue measure. In case m = 2 and 7r = (12----~ 21), the (A, ~r) interval exchange is simply addition by A2 modulo I A I. It is well known that minimality, which here is equivalent to the irrationality of A2/IA I, implies unique ergodicity. When m = 3, there is only one permutation, ~" = (123--~ 321), such that ~ is irreducible and (A, 7r) has 2 points of discontinuity. In this case, Keane notes that minimality is equivalent to the irrationality of (IAI-A3)/(IAI-A1), and again minimality implies unique ergodicity. The extremals of E(A, It) are well known to be the ergodic invariant measures, and any two ergodic invariant measures are either mutually singular or multiples of one another. Keane, extending a result of Oseledets [13], proves a minimal interval exchange has at most m + 1 pairwise mutually singular ergodic measures, and notes his proof can be altered to obtain the bound m. As ~(A, or) is spanned by its 224 w.A. VEECH extremals, dim 2(A, or) =< m. Notice that because any collection of nonzero, pairwise mutually singular measures is linearly independent over R, (0.4) v(A, ~r) = dim X(A, Ir) is the number of ergodic invariant measures for (A, or). In Section 2 we shall associate an m x m matrix (of O's, and • l's) L" to r and prove that dimE(A, or)_-<1 rank(L') if (A, It) is minimal. 0.5. Theorem. I[ (A, lr) is a minimal interval exchange, then 1 (0.6) v(A, ~r)<~ rank(L'). In particular, v(A, r < [m/2]. Keane conjectured that minimality implies unique ergodicity, as apparently did a number of mathematicians out of print, but Keynes-Newton [11] produced examples with m = 5 and v(A, r 2. In turn, to negate a conjecture by Keynes-Newton, Keane [9] gave examples with m = 4, v = 2. By (0.6) and the remark following, these examples maximize v(A, or) for m = 4,5. Keane's construction employed the ergodic theoretic device of inducing a transformation on a subset (this will be defined below), and as was stated in [15], his procedure suggested a formal study of this process as a possible way to settle his conjecture 0.7. Conjecture (Keane [9]). If ~r E ~m is irreducible, then for almost all A E Am the (A, 7r) interval exchange is uniquely ergodic. The purpose of this paper is to begin such a formal study of the inducing process. Our ultimate hope is that this will not only lead to a solution to Keane's conjecture but, more importantly, to a usable criterion for the unique ergodicity of (A, or). This could be of use, for example, in the study of "rational billiards" (see [10]). We shall define a set ~+~ _C ~,. and show that if ~r E ~, and if (A, It) satisfies the i.d.o.c., then for a certain collection of "admissible" intervals [~, 1,/] C_ [0, IA I), the INTERVAL EXCHANGE TRANSFORMATIONS 225 induced or first return time transformation is a (translate of) [A ct), 11"1) interval exchange, I Atl)l = ~ - ~, rr~ ~ @~, and there exists a nonnegative integer matrix A, det A = --- 1, such that A = AA t" (compare with [9]). Repeating this procedure, i.e. using a decreasing sequence F~ D F2 D 9 9 9of such admissible intervals, one obtains an "expansion" (0.8) A = A (~ (27... A (")A (") and it will develop that if F, ~O or a point (0.9) S(A, rr) = f'l A (''" "A<")A,- n is naturally isomorphic with X(A, ~r). In particular, if there is a matrix B with positive entries such that B occurs infinitely often as a product A (')A 0+~)... A o) in (0.8), then (0.9) is one dimensional, and (A, 70 is uniquely ergodic. It is in this fact that one might expect to find a usable criterion for unique ergodicity. One may specialize F,, setting F1 = [0, A~) and in general F,+I = [0, A]"~). Here one obtains a map if0:~2 • Am--~ ~2 x A= and an associated "projective" map ~:~+Xam_X'-*~++Z~m_,, Am-l={~, EArn [[,A[= 1}, where f AO) '/7"1) An application of the Poincar6 recurrence theorem, in conjunction with the criterion of the preceding paragraph will show that if ~r possesses a finite invariant measure equivalent to Lebesgue measure, then Keane's conjecture is true. In case m = 2, ~ = {(12---~ 21)}, there is an infinite invariant measure (viewing A~ as (0, 1), it is dx/(1-x)). In case m ---3, ~ = {(123---~321)}, we are able, after a rather lengthy argument, to assert that there is an invariant measure equivalent to Lebesgue, but it remains to be seen whether it is finite or infinite. In any case, we do obtain that if B~, 9 "" , Bk are matrices which can occur in (0.8) in the case of ~*3~ then for almost all A, the block B1B2. 9 9B~, occurs infinitely often as A (,§ A 0+~) in (0.8). 226 w.A. VEECH 01. The cone of invariant measures Fix m > 1, (A, 7r) E ~,, x Am, and assume (A, 7r) is minimal. As before ~(A, 7r) is the cone of nonnegative nonzero T invariant Borel measures, T = Tr We shall in this section identify ~(A, 7r) with a cone S(A, 7r)C_ Am. Let/z E E(A, 7r). Being minimal, T can have no periodic points, and therefore/z has no atoms. Also, if ~7~ O is an open set, /z(~7)>O. It follows that ~(x)= ~[0,x), 0~x<l;~l, is continuous and one-to-one. Define )t (tt)E A,. by x, (g) = g (/3,_,(x ). /3, (x)) = ~ (/3,) - ,p~ 03,-1). Next, define T. on [0, [A (g)l) by (1.1) T,,p~ (x)= ,p, (Tx). T. is, as we shall soon prove, the (A (/z), zr) interval exchange. Notice that by (1.1) (1.2) T~o,(x)= ~o,(T"x) (n EZ). 1.3. Lemma. With notations as above T~ is the (A (tt), or) interval exchange. ProoL If x E [13~-~(A),/3~ (A)), then by definition Tx = x -/3,_1(A) +/3~,_,(A"), and ,p,.(Tx) = ~[0,x - t3,-,(X) + t3~,-,(X ~)) = ~.L [0, ~Iri--l(/~ ~r)) "4- [.~ [~r,--l(/~ ~r), ~Tri--l(/~ "n') "JI- X -- ~i--l(/~)). Now use the invariance of/z for both terms on the right to obtain ~ (ix) = t3~,_l(x (~)~) + ~ [~,-,(x), x) (1.4) = 9, (x)- t3,_,(x (~)) + t3,-dx (~)").
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