LEAVES OF MARKOV LOCAL MINIMAL SETS IN FOLIATIONS OF CODIMENSION ONE

JOHN CANTWELL, LAWRENCE CONLON The authors continue their study of exceptional local minimal sets with holonomy modeled on symbolic dynamics (called Markov LMS [CC 1]) . Here, an unpublished theorem of G . Duminy, on the topology of semiproper exceptional leaves, is extended to every leaf, semiproper or not, of a Markov LMS . Other topological results on these leaves are also obtained .


Introduction
Let M be a compact, orientable manifold, ,F a transversely orientable CZ foliation of M of codimension one .Each component of 8M, if there are any, is to be a leaf.
Let X be a local minimal set (LMS) of F. That is, there is an open, connected, .P-saturated subset U C_ M and X is a minimal set of FlU.A special class of such sets, called Markov LMS, was introduced in [C-C 1] .These are exceptional LMS such that the holonomy pseudogroup of the foliated set (X, .FIX ) is, in a certain sense, generated by a subshift of finite type (see §1) .We will see, in §6, that every such subshift can occur for suitable Markov minimal sets in suitable C°°-foliated 3-manifolds .
Let E(L) denote the set of ends of a leaf L C X, a compact, totally disconnected metrizable space of ideal points of L at infinity.Let E*(L) denote the closed subspace of ends that are asymptotic to L. In case X is a minimal set of F (i .e., we take U = M), it is clear that E(L) = E*(L).
The first author was partially supported by N .S .F. Contract DMS -8420322, the second by N .S .F .Contract DMS -8420956 .The second author would also like to thank the Centre de Recerca Matématica de Institut d'Estudis Catalans, Barcelona, under whose hospitality this paper was completed .
Theorem 1. Le¡ X C M be a Markov LMS of .F and let L be any leaf of .FIX .Then £*(L) is homeomorphic to the Cantor set.
G .Duminy has proven Chis for semiproper leaves in an arbitrary exceptional LMS (unpublished) .It is unknown whether his result extends to all leaves of the LMS .
Definition.A leaf L is resilient if it has a holonomy contraction and if L itself is captured by this contraction .
Definition.A handle in L is a compact; connected, nonseparating submanifold H of codimension one, áH = 01.The genus of L is the maximal number of pairwise disjoint handles in L that are linearly independent in H* (L ; R) .
Theorem 2. Le¡ X C M be a Markov LMS of .F. Then X contains exactly a countable infinity of resilient leaves .Furthermore, either genus(L) = 1, for each resilient leaf L C Jr', the remaining leaves of .FAX having genus 0, or every leaf L C X has infinity genus and every end E E £*(L) is a cluster point of handles.
It was shown in [C-C 1] that each semiproper leaf in X is resilient, but that there are only finitely many such leaves .
If .1'C M is an exc_ eptional LMS for .F, fix T C R, a finite union of open, bounded intervals, let T denote the closure of T, a compact one _ manifold, and fix an imbedding p : T -> U, transverse to F and such that C = X fl p(T) is a Cantor set .Here, of course, U is an open, connected, .F-saturated subset of M such that X is a minimal set for .FIU .Let I' denote the pseudogroup on T induced by the holonomy of the foliation .F and let I'IC be the induced pseudogroup on C .H'y an abuse, we suppress all mention of p hereafter .Definition.Suppose that the above choices can be made so that there is a subshift of finite type o : 1C p -> Kp and maps r : C -+ C and : C -> Kp having the following properties : (1) T is locally a homeomorphism and I'IC is generated by the local one-one restrictions of r ; (2) 77 is a continuous surjection; (3) rl -1 (97(x)) is either a singleton or the pair of endpoints of a gap of C, for every x E C ; (4) 17or=uor7 .Then T is said to be essentially a subshift of finite type generating FIC, C is called a Markov I'-minimal set, and X is called a Markov LMS of P.
The constructions of Sacksteder [Sa 1], Raymond [Ra], Hector [He], Ghys and Sergiescu [G-S], Inaba [In], and others provide examples of Markov minimal sets of a foliation P. It is not hard to modify these to produce Markov LMS that are not minimal sets of .P.
Various elementary properties of Markov LMS were treated in [C-C 1, §1].For the sake of completeness, we review these briefly.
For x E R(hi), set t(x) = h, 1 (x) E D(hi) .This well defines locally a homeomorphism, such that t(ZO) C Zo .Set rS = t i Zo : Zo -) Zo > a continucus map that is locally a homeomorphism, and remark that I'SIZO is generated by Ts .
Let ¿ = (i,,)°°_ 1 E lCp, and let wn be defined inductively as wn _1 o hi .Then Iw+1 C Iw and the set I, = n°°_ 1 I, is either a singleton or a nondegenerate, compact interval .Let I, denote the set-theoretic boundary of I, , a set with one or two elements.It is clear that Z = U¿EKP I¿ and Zo = U¿EKP I¿ .Define i7s : Zo -Kp by rls(ij = t and remark that rls o -rs = o-o 91s.Thus, we can say that Ts is essentially a subshift of finite type. 1 .1 .Lemma.The set Kp is a Cantor set and ro-minimal if and only if there exists a rS -minimal Cantor set C C_ Zo such that g1s(C) = Kp .In this case Z o \ C is a union of at most countably many rs-orbits, each of which accumulates exactly on C.
For the elementary proof, see [C-C 1, (1 .1)] .If, in fact, rs is of class at least C2 , it is true that Zo \ C is a union of at most finitely many rs-orbits, but Chis follows from a much more difficult theorem [C-C 1, (6.1)] .
1 .2 .Lemma.Let X be a Markov LMS of Pand let C, r, P, Kp, T, and rl be as in the definition of such a set.Then this data can be chosen so that there exists a Markov sub-pseudogroup rs C_ r, defined relative to the matrix P, such that C C Zo, rlsIC = al : C -> Kp, TsIC = T, and rsIC = FIC .This was essentially proven in [C-C 1, (1 .2)] .The relatively minor adjustments that are needed for the above formulation are left as an exercise.
Let X C U be a Markov LMS .Thus C = X n T is a Markov Ir-minimal set .
We fix a Markov sub-pseudogroup rs C r as in (1.2) .
Let x E C and let Irs(x)i denote the digraph of the orbit rs(x) relative to the set {h1, . . ., hz } of generators.Here, each positively oriented edge denotes an application of some hi, 1 _< i <_ ni, hence lnoving backward along an edge denotes an application of T. By elementary symbolic dynamics, it is easy to see that this graph contains at most one cycle and that, if v E IF(x)l is a vertex not on a cycle, every positively directed edgepath out of v of infinite length meets no cycle.Moreover, infinitely many vertices of such an edgepath are"branch points", in the sense that "at least two distinct, positively directed, infinitely long edgepaths of 1rs(x)1 emerge from each of them.These observations have certain easy consequences . 1 .3 .Lemma.The space £(Irs(x)I) of ends of the graph JIrs(x)I is a Cantor set.
We will prove Theorem 1 by showing that, under the hypotheses of that theorem, if Lz is the leaf of XIX passing through x E T fl X = C, then there is a homeomorphism £(Irs(x)I) -£*(Lx) .Our proof of Theorem 2 will show that the above holonomy contractions are compactly supported .Such a contraction determines a compactly supported cocycle on the leaf L that is nontrivial in H*(L ; Z) .That is, if the graph irs(x)j has a cycle, that cycle is Poincaré dual to a handle on the leaf through x.
Remark that so far in this section we have not needed smoothness hypotheses.For the following, smoothness of cases at least C2 seems to be necessary.1 .6 .Lemma.If F is of cases C2 and X C M is a Markov LMS, then the Markov system S of (1 .2) can be chosen so that I, is a singleton, for each c E Kp .In particular, 77 : C , Kp is a horneomorphism.
Proof.. We modify the Markov system S of (1 .2) .If I¿ is a nondegenerate interval, then [C-C 1,(3.4)]implies that t = (Jo, J1 , J1 . . . ., J1 . . . .), where Jo and J1 are finite sequences of elements of {l, 2, . ..,ni} .By applying a suitable nonnegative power of r to I,, we ma.y assume that Jo = 0 .We set J1 = (j1,j2, . ..,j9) .If Iik and T k-1 (Ij share an endpoint x, delete {x} U r k-1 (int(IJ) from IB A, .Otherwise, delete the interval rk-1 (int(IJ) from Ii, Similarly, delete {r(x)} UT k (int(IJ) or rk'(int(I,)) from XjA, .These operations should be performed for 1 _< k <_ q .This breaks each Ij,_ and Xik into a finite number o£ subintervals I1,`1 , . ., I7k ?nk and X1A, 1 , . . ., Xjk n,k (remarle that possibly jk = jj, k l) .The generator hj,_ of I'S is then broken into hjk i X;ki ---, Íjki, 1 < i < " 1 r., 1 < k _< q.It is elementary to check that this modification of S is again a. Markov system for the Markov LMS X .Finally, again because of the C2 hypothesis, there are only finitely many rs-orbits in Zo that are semiproper [C-C 1, (6.1)], so the above procedure only needs to be iepeated finitely often to produce the desired Markov system . 1 .7 .Corollary.If P is of class C2 , if -X C M is a Markov LMS, and if e > 0 is given, then ¡he Markov system S of (1 .,2)can be modified to a Markov sysiem S*, for the same LMS 1X, such that ¡he intervals X; and Ii are all of length less than c and each generator h; is the restráction of a suitable generator hi(i) from S.

. The combinatorial arguments .
The dynamical description of .Pis given via a suitable choice of open cover {Uj,,E .A of M by Frobenius coordinate charts .Each Ua is to lie in the interior of some Frobenius chart a.nd an .'F-plaque in Ua is to meet at most one .P-plaque in Up, for all a, 0 E A .
The set of (closed) P-plaques in Ua can be identified with a compact, imbedded, P-transverse arc Ra > M, the preimage Ra being, itself, a compact subinterval of R .M7e arrange that these subintervals be disjoint in R with disjoint images in M. lf Pa E Ra and Pp E Rp are interpreted as closed plaques, we will write 713a(Pa) = P/3 if and only if Pp and Pp overlap in the sense that int(Pa ) n int(Pp) qÉ 0.
This defines the set of generators Gw = {ypa}a,pEA of the holonomy pseudogroup rp of F, a pseudogroup on the compact one-malifold RF = U,,EA R, Thus, a word of length n in these generators, when applied to a plaque P E RF, amounts to a cha.in (P = PO , P,, . . ., Pn) of plaques such that P2_ 1 and P4 overlap,1<i<n .
Let X be a Markov LMS of P. We can assume that the imbedding T --> U, relative to which X is of Ma.rkov type, is the imbedding of a compact submanifold T C RF induced by Rw > Dil .
By (1 .2),these is a. set Gs = {hi},< ;< ., generating a Markov sub-pseudo group r, C FF , such that FFIC = rsrC .Each hi is locally a composition of ya p's .By (1 .7),we can brea.keach hi up into finitely many of its restrictions to disjoint, open subintervals of D(hi ), on each of which hi is a pure composition óf yap's.Thus, wlog, -,ve assume that each lii is a reduced word in the generators Gs .
We must take into a.ccount the structure of the foliated manifold (U, .P) obtained by completing (U, .P1U) relative to any Riemannian metric inherited from be a decomposition into a compact nueleus K (a foliated manifold with convex corners) and arms Vi (foliated interval bundles), as déscribed in [Di] (also, see [Go]).Let B C_ A be the subset such that Ua C int(K) if and only if a E 8. Let R = UaEG R« .
The nueleus can be chosen large enough to engulf any specified compact subset of U .Thus, without loss of generality, we can assume that the Frobenius cover {Ua}aE .4 has been chosen so that T C_ R and the expressions for the generators hi of I'S as words in the generators yaa of F.,F involve only indices a Q E 13.Let G = {yap}a,QEti and let I' be the pseudogroup on R generated by the set G. Thus, I's 1C = I'1C.
Definition.The foliated manifold _ (M', Y) is defined by setting M' _ U-EB Ua and Y = FIM' .The subset X' C M' is the .r'-saturation of C .Remark.Even if the foliated manifold (M, .F ) has some degree of smoothness, aM' is only piecewise smooth, being divided by corners of various descriptions into smooth pieces that are each either transverse or tangent to .'F' .
In light of our discussion so far, the following is an exercise.
2.1.Lemma.The set X' is an exceptional minimal set of Y.Each leaf L' of .F'lX' is contained in a unique leaf L of .FIX and the correspondence L +--> L' is one-one between the leaves of FAX and of .T''jX' .
For b E Ci and x E Rb fl X', we can chosee y,; E I' to be a word of shortest length in the elements of G such that -y,,: (X) E C. By compactness of R6 and of R6 fl X', there are finitely many compact subintervals with interiors covering int(R6), each meeting X' in a set that lies entirely in the domain of some such yx and is carried by yx into C .That is, as ó ranges over Ci and x ranges over R6 fl X', we can arrange that only finitely many yx are distinct .Remark that, if x E C, then yx is an identity.
Finally, whenever a Q E Ci and x E D(yo) fl A", we can write where yy and yx are as above and w .0 is a reduced word in the generators Gs.Again, by compactness, there will only be finitely many distinct words wa, that occur as a and /3 range over 8 and x ranges over D(yaR) .Definition .For w E I's a pure reduced word in the generators Gs, the length of that word is llwils .Similarly, for a pure reduced word -Y E I' in the generators G, the length is denoted by ¡¡y¡¡ .
It is cleax from the above discussion that we can fix a choice of N E Z+ that is simultaneously an upper bound to each of the following Remark also that this implies (d) Ilwápll <_ N2 .Let L be a leaf of FIX, x E L fl T, and let L' be the corresponding leaf of F'IX' as in (2.1) .Then the graphs Ir(x)I and II's(x)I will be defined relative to the, generators G and Gs, respectively, and so will be quite different .Also, since y&, = y0,á, we will agree that both 7p,,, and yapare to label the same edge with opposite directions .By contrast, each edge of II's(x)I has been given a unique label hjand a corresponding preferred direction (cf.§1).
The graph Ir(x)I is the 1-skeleton of the nerve of the plaque-cover of L', hence the first of the following lemmas is evident .Proof.For a, 0 E G, let z E L n D(yap) and let y = ya,9(z).Recall that where IIy.II, Il yy' II, and IIwáp11s a.re all bounded above by the integer N and IIwáj is bounded above by N2 .
The vertices of Irs(x)I are contained in the set of vertices of Ir(x)I and this inclusion extends to a mapping Irs(x)I -+ ¡F(x)l, not of graphs but of topological spaces .Indeed, an edge of Irs(x)I labeled by hj is carried onto an edgepath in IF(x)I corresponding to the expression for hj in terms of the generating set G.
The above mapping induces a : £(II's(x)I) -> £(IF(x)I), a continuous map.This is a. surjection since every vertex of IF(x) I can be joined to á vertex of II's(x)I by an edgepath in IF(x)I of length at most N.
We must show that A is one-one .Let e and e' be distinct elements of £(Irs(x)I) .Let {xk}' 1 and {x'}~1 be sequences of vertices of Irs(x)I con-k= to e and e' respectively .Then, there is a verter x* of this graph such that, for k sufficiently large, any edgepath in II's(x)I joining xk and x' must pass through x* .Indeed, a: * will be one of the points at which the graph "branches" .
Let D(x* , v) C IF(x)I be the set of vertices that can be joined to x* by an edgepath of this graph of length at most v.In order to show that A(e) :~A (e'), it will be sufñcient to find an integer v > 0 such that, for k sufficiently large, every edgepath in IF(x)I joining x,. and x' must meet D(x *, v) .For simplicity of notation, let z = xk and z' = x' .An edgepath joining these points corresponds to a word y = y6 y ay_ 1 0 " -O yalap such that y(z) = z' .Let z(0) = z and z(p) = -Yavav-i p . . .p ^Yalao(z), 1 <_ p < q .Since z E C and z' = z(q) E C, we see that 7, and yz, are identities, hence that y(z) = w(z) ='tva9á4r) o . . .o w z(ea)o( z) .
Here, each wa(,á.i) is a word in the generators Gs, hence the above expression represents an edgepath in irs(x)j joining z a.nd z' and, as such, passes through x, .Let us say that wáná y i) j represents a segment meeting x* .It follows that z(p) E D(x, NZ + N) .
Since endsets are compact Hausdorf spaces, we conclude that A is a homeomorphism.

. Corollary . If X is a Markov minimal set of F, then the conclusion of
Theorem 1 is true .
2.6 .Lemma.If X C M is a Markov LMS of F, then X contains exactly a countable infinity of resilient leaves .
2 .7 .Lemma.Leí X C 117 be a Markov LMS of F, leí L C X be a resilient leaf, and let x E L fl T C C. Then, the holonomy coníraction that represents the generator of h,; (L', X') = 7Í (L, X) -Z is compactly supported on L' .
Proof. .We can a.ssume that a, E Irs(x)j lies on the unique cycle w of that graph and we view w as an edgeloop based at x .let D(w, v) C IP(x)j denote the set of vertices that can be joined to a. vertex of w by an edgepath in this It is a consequence of Theorem 3 that, for a suitable (open) normal neighborhood W of F in U, each component F of Jr' fl W is diffeomorphic to F,, in such a way tha _ t the projection p : F --> F, defined by projection along the normal fibers in W, is identified with 7r : F,,.-> F.Here W will be diffeomorphic to the manifold Q which is obtained from Fq x [x, b] by the identification map The manifold Q is naturally foliated by leaves diffeomorphic to F,,, asid the natural projection p : Q --+ F restricts to 7r on each of these leaves.The diffeomorphism W -Q carries each component of X fl W opto one of these leaves and identifies p with projection along the normal fibers of W. This picture and the proof are, by now, standard in geometric foliation theory.For example, see [C-C 2, §6] for the completely analogous situation of totally proper leaves winding in on leaves at lower levels .The situation there is precisely the one described here, but with X replaced by a single leaf.
Let Fi, . . ., Fq be the components of aU on which X accumulates, let Hl , . . ., Hq be the respective handles, and let VVi, . . ., Wq be the corresponding normal neighborhoodsjust described .Let F q _ +1, . . ., Fp be the remaining components of aU _ a n d Wq+1, . . ., 4Vp respective normal neighborhoods of these in U that do not meet X .We can a.sstune that, for q + 1 < j <_ p, Wi fl K is a finite union of closed Frobenius charts for F. Fina.lly, let M' = K \ U,<i<p Wi.By Theorem 3, FIX has no holonomy outside of M' .At this point, the following is clear .
3.1 .Claim.The Frobenius cover {Ua},EA for the foliated manifold (M, .F) can be chosen so that the manifold M', as described aboye, coincides with the manifold M' construcied in §2 .
As in §2, we set .F' = .F1AP, a foliation with exceptional minimal set X' .Let L be a leaf of FIX and let L' be the unique leaf of .F'IX' contained in L (2 .1) .Because the foliated manifold (M',,F') has been fashioned with greater care than in §2, the leaf L' is a manifold with boundary, but with no corners .Indeed, (M', F) is a. foliated manifold with corners of various descriptons, and these corners divide OM', as usual, into a part tangent to .F, denoted a,M', and a parí transverse to F, denoted afiM'.The components of aL' are exactly the components of L fl BnhM', each being a copy either of Hi, 1 < i <_ q, or of aBj , 1 <_ j < r .There are infinitely many such components.Finally, each component of L \ int(L') has as boundary exactly one component of aL' and either lies entirely in T4ji, 1 < i < q, or in some V 1 < j < r.Proo£ Since £(L') = £ * (L'), the image of A is contained in £*(L) .We must show that A maps £(L') one-one onto £*(L).
Let e and e' be distinct ends of L'.Let ~k C int(L') be a compact subset separating e and e' .Since the components of L \ int(L') correspond one-one to the components of áL', it is clear that 0, as a compact subset of L, also separates \(e) and A(e'), hence these ends are distinct in £*(L) .
Let e E £*(L) .Let . . .C Uti C Uk_ 1 C . . .C L71 C L be a fundamental system of neighborhoods of e .Since (b 9É Uk f1 1V1' = Uk (1 L' (the second equality is a consequence of (2 .1))we can choose xk E Uk fl L'.By passing to a subsequence, if necessary, we can assume that the sequence {xk}k 1 converges to an end rt E £(L') .It follows that A(17) = e.
Proof.As usual, the contraction on L defines a nontrivial cohomology class 0 E H 1 (L ; 7L) .If we view this class as a homomorphism 3 .5.Corollary .Theorem 2 is true .0 : 7r, (L, ti) -+ 7L, we see, by the geometric consequences of Theorem 3, that it va.nishes on any loop that is freely homotopic to a loop on L -, int(L').Thus, 0 is supported in int(L') where, by (2 .7), it is compactly supported .At this point, the proof of (2.8) applies without change .

. The fundamental contraction .
Let X be an exceptional LMS, not necessarily of Markov type, and let F be any proper or semiproper leaf of the foliation F. By transverse orientability, F has two sides, denoted F+ and F-.Assume that X is asymptotic to F on at least one side, say on F+ .Let 0 E F a.nd let [0, e] be a parametrized, .F-transverse arc, issuing from 0 on the side F+ As in §3, h o denotes holonomy in [0, e] defined by a loop o on F, based at 0, and o * H denotes the homologica.lintersection number of o with a handle H .
Definition .Let a : [0,1] ---+ L be a.loop based at o(0) = o(1) E H.If o¡ ]0,1[ is in general position with respect to H, if o¡ ]0, b] and o¡ [1 -b, l[ both lie on the lame side of H for small b, and if o always crosses H in the same direction, then we will say that a is in normal position with respect to H .

. Theorem .(Duminy)
. If e > 0 is sufáciently small, then there is a handle H C F and a loop u on F, based at 0, such that o, * H = 1 and ho = f is a contraction of [0, e] to 0. Furihermore, for some integer r£ >_ 0 and every loop T on F, based at 0 and in normal position with respect to H, Remark.If F happens to be a semiproper leaf that lies in X itself and if F+ is the nonproper side of F, this theorem leads rather easily to the result that £*(F) contains no isolated points, hence is a Cantor set.
Let U be an open, connected, .F-sa.t_urated set such that X is a miniimal set of FI U. Let F be a.component of r7U on which X accumulates in U .Our main application of (4.1) will be to this situation, the sitie F+ being the sitie bordered by L1 , so F+ will be a.proper sitie of F. The conclusion of (4.1) will be our starting point for the proof of Theorem 3 .
Unfortunately, Duminy has never made his proof of (4.1) available to the mathematical public .The result is crucial for current research into the structure of exceptional minimal sets, so we have prepared an account [C-C 3] for informal circulation .
Definition.The handle H C F, given by (4.1), will be called a holonomy handle .

The proof of Theorem 3
As in §3, the decomposition U = K U Vi U . . .U_1;. induces a decomposition F = AUBj, U. . .UBj,, where F is a component of DU .13 y suitably renumbering, we take Bj; = Bi, 1 < i < t.
Let xi E Bi a.nd let Ji be the fiber over xi of the interval bundle Vi --~Bi, 1 <_ i <_ t.Let Po denote a.n F-plaque containing the point 0 E A. Edgeloops on jFw(0)j, based at 0, induce a holonomy sub-pseudogroup rF C r y, defined on open neighborhoods of 0 in [0, e[ and fixing 0. The usual holonomy group xo(F) is the group of germs at 0 of the elements of FF .We say that a chain p = (Po , P,, . . ., Pf ) of pla.ques, without repetitions, is a simple chain a,t Po, as is the holonomy element h,, E F_F that it induces .A chain u = (Po , P,, . . ., Pq , Po ) and the associated ho E rF will be called a simple loop at Po if (Po , P,, . .., Pq ) is a simple chain.Finally, if p = (Po, . . ., Pq ) is a simple chain and a = (P9 , Pq+1, . . ., Pq+9, Pq) is a simple loop at Pq , then and T=PuP =(Po, . ..,Pq,Pq+i, . ..,Pq+9,Pq, . ..,Po) h r =hp1 oh o oh p EFF are called basic loops a.t Po.Every element of FF, restricted to a suitable neighborhood of 0 in [0, e[, can be written a.s a composition of basic loops at Po .Thus, the (gerins of) basic loops generate ho(F) .
Let A = A o C_ A 1 C_ . . .C_ Ak C_ ---C F be ara exhaustion of F by compact, connected submanifolds with boundary .These should be chosen so that the componente of F\ Ak are not relatively compact in F arad so that these components correspond one-one to the componente of c9Ak .Let G o denote the set of basic loops at Po which, as plaque chairas, consist entirely of plaques meeting A o .Inductively, suppose that G k has been defined, some k >_ 0. Then Gk+1 \ Gk is to be the set of basic loops of the form hp 1 o hQ o hp , where p is a simple chaira at Po that involves only plaques that meet Ak and contains just one plague that meets egAj, 1 <_ j < k, a.nd o, is a basic loop that involves only plaques that meet Ak+1 \ Ak View each Gk as a subset of FF and let Go = U'0 Gk C FF .
5 .1.Lemma.For e > 0 sufccientlg small, and for each g E GI, either g or g-1 is defined ora [0, e] and maps that interval into itself.Furihermore, Gp generates ho (F).
Proof. .Since Go is finite, choose e > 0 so slnall that both g and g-1 are defined ora [0, e] with images in [0, e -{-b[, for some b > 0 and each g E Go .
Either g(e) or g-1 (e) E [0, e] .Making e smaller, if necessary, we make sure that hp is defined ora [0, e], for each simple chaira p a.t Po in A .Since Vi is a foliated interval bundle over Bi, 1 < i _< t, it is true that either g or g -1 sends [0, e] to itself, for each g E G t \ Go_ The fundamental group 7r, (F, 0) can be defined via the nerve of the plaque cover of F (provided the Frobenius cover of 1V1 has been suitably chosen) .Clearly the set of edgeloops in 1FF(0)1, corresponding to Go, generates 7r1(Ao,0) .By the Van Kampen theorem and induction ora k, the set of edgeloops, corresponding to Gk, generates 1r 1 (Ak, 0), hence those corresponding to G tt generate 7r 1 (F,0) .The natural surjection 7 1 (F,0) -> I-to(F) is then used to prove that GI generates Ho(F) .
Definition .Let g E Gp \ C=k .The k-representation of g is hp 1 o hQ o hp, where p is a simple chaira in Ak and o-is a. basic loop involving only plaques that meet F \ Ak .
Such a k-representation always existe a.nd is unique .Its usefulness lies in the fact that, while Gq \ Gk is generally ara infinite set, the k-representations involve only finitely many of the simple chairas hp .5.3.Lemma.If e > 0 and.ó > 0 are suitably chosen, each as small as desired, then every g E Crq defines g : [0, e] -> [0, ¬ + 6[.Furthermore, given 0 > 0, k can be chosen so large that jg(u) -uj < 17, for each g E Gl \ Gk, 0<u<e .
Since Gk is finite, we can choose e E ]0, e'/2[ so small that g sends [0, e] into [0, e'], for each g E Gk, hence for each g E Gp .Take 6 E ]e' -e, e'[ and obtain all assertions .5.4.Corollary .Le¡ 0 < r < 1 < s.Then e and 6 can be chosen as in (5 .3)and k > 0 can be chosen so that, for all g E Gtt \ Gk and for 0 <_ u <_ e, ¡he inequalities r < g'(u) < s hold.If, furthermore, {g 1 , . . ., gn } C Gk is a subset such that no gi is germinally equivalent lo a contraction lo 0, then ¡he choice of e can be made so small that r < gi(u) < s, 1 < i < n, 0 < u < e.
Proof.-By (5.2), we choose k > 0 such that, for g E Gt \ Gk with krepresentation g = h. 1 o ho o hp, we have 0 < u < e.By elementary calculus, 0 <_ u < e, and each of the finitely many functions h' is continuous .By making e (and, if desired, b) small enough, we gúarantee that 0 < u < e, hence hv( u ) ~< hp(g(~)) C ' r < g 1 (u) < s, 0 _< u <_ e.There remain the elements gi E G k , 1 <_ i <_ 71 .Since none of there restricts to a contraction to 0 on any neighborhood of 0 in [0, e], we see that gi(0) = 1, 1 <_ i <_ n.Thus, by making e > 0 possibly smaller, we complete the proof At this point, we consider a Markov L_MS, _X C M, assumed to be a minimal set of Fl U and to accumulate on F C aU in U. We let ]a, b[ be a component of ]0, e[ \ X, as in §4, and consider the holonomy handle H C F a.nd the holonomy contraction f = ho given by (4 .1) .For each integer n >_ 0 and for u E D(f), we set un = fn(u) .5 .5 .Claim.Without loss of generality, we can assume that the .'F-transverse1-nianifold T, as in §1, is ]al , bo[ .This implies that C C [b1,ao] .Proof.. (1) We construct a holonomy imbedding 0 : C --+ ]a l , bo [ If this has image X n ]a l , bo [, the first assertion follows.VVe will consider the contrary possibility in step (2) .
From the above paragraph and elementary symbolic dynamics, it follows that the points y E C such that 77(y) is a periodic point in {1, . . .,nz}' are dense in C n Iti, 1 _< k <_ 7n .Since only finitely many periodic points can pertain to a semiproper I'-orbit [C-C 1, (6.2)], we choose yA E C n Ik such that C clusters on yk from both sides and 7)(yti) is periodic.Then there is a holonomy contraction 9R of Ik-to yk .Let cpti be an element of holonomy of .FIU that carries yti to x,k E ]a l , bo [ .It is easy to arrange that x 1 , . . ., x,k all be distinct.For a suitably large integer 7a, Ok = cp 1 , o 0 is defined on all of Ik., 1 <_ k _< m, and {Ok(IR)}k1 is a set of disjoint subintervals of (2) We now assume that C C ]al, bo[ .If K = X n]al,bo[ \ C Y 0, then K is a Cantor set .For each a, E K, choose y, as in §2, a holonomy element with y,;(x) E C .As remarked in §2, we can arrange that only finitely many y,, be distinct .Thus, {R(y,)},El< is actua.llya finite set of open intervals.
By (1 .7),we can assume that the Markov system S is such that, for each x E K and 1 < i < 7n, either Xi n R(-y,;) = 0 or Xi C R(y, ) .Select an open interval U, about each Xi with these same relations to the intervals R(yx) .One then enlarges S by adding the generators -y, 1 jUi and the intervals y.'(Xi) .
This Markov system generates the Cantor set Jr' n ]a l , bo [, so we can take We can assume that H C int(A), hence by cutting A along H we produce a compact, connected ma.nifold A' .It is fairly obvious how to select a subset of basic loops in A that should be thought of as the basic loops on A' .Let Gó = {g1, . . ., g.m} C Go be the elements of holonomy corresponding to the basic loops in A' .Then, if k >_ 0 is sufficiently large, the set {f } U Gó generates the same holonomy on [0, bti,] as does Go .By renumbering, replace bo with bk .Thus, if Gy = Gó U G1 U . . .U Gs U ---, we can assume that Gt = Gá U {f }.
In order to prove Theoreln 3, it should be clear that proving the following two propositions will suffice .5.6.Proposition .For k >_ 0 sufficiently large, Gy ftixes every point of X n [0, bti ] .5.7.Proposition .It is possible to choose ¡he decomposition of U so that the total holonomy of the foláated interval bundle Ji ---> Vi -----> Bi faxes every point of 11' n Ji, 1 < 2 < t.
Proof.. (Using (5 .6))By [C-C 1, Theorem 1], each of the semiproper leaves L C X has a holonomy contraction on its nonproper side, say on L+ , and this contraction is unidue relative to X. Apply (4.1) to the side L+ so as to conclude that this contraction is compactly supported .By [C-C 1, Theorem 2], there are only fi_nitely many semiproper leaves in X, so we choose the customaxy decomposition U = K U Vi U . . .U V,. in such a way that K engulfs the holonomy handle on each of them.
Parametrize J; a.s [0,1], 1 <_ i < t.By (5 .6), the loops on Bi, based at xi, fix every point of X n [0, b], for some 6 E ]0,1] .The subset of X n ]0,1[, fixed by these loops, is closed in ]0,1[, hence, if it is not all of X n ]0,1[, we can choose the above 6 E X n ]0,1[ in such a way that some loop on Bi at xi induces a germinallyy nontrivial element of holonomy on [b, b + e[ n Jr' that fixes 6 .Again appealing to [C-C 1, Theorem 1], we see that, for e > 0 sufficiently small, the holonomy around this loop gives a contraction of [S, 6 +e[ to 6.
But this loop also fixes each point of [0, S] fl X and, in a Markov LMS, each holonomy contraction to b contra,cts a whole neighborhood of 6 in ]0,1[ fl X to b (elementary symbolic dynamics) .Thus, ó must lie on a semiproper leaf and this contradicts the fact that the holonomy handle on that leaf does not meet vi .
In order to prove (5.6), we first review certain poüzts from [C-C 1, §6], where the finiteness of the number of semiproper leaves in X was established .
By the parametrization of T as the subinterval ]a l , bo [ C R, we view I'S as a pseudogroup on R and place the following definition.
Definition .A point x E R is rs-unifoim if there is an open neighborhood V of x in R and a number v > 0 such that g'(u)/g'(v) < v, for each g E rs a.nd for all u, v E V fl D(g) .The set of rs-uniform points is denoted by U+ and its complement in R by B+ .
Remark .Since U' is open in R and, by default, contains R \ ]al, bo[, the set Ci+ is compact.
In the following, T again denotes the essential subshift that generates I'IC.A T-cycle is a subset {x,T(x), . ..1Tk(x)}C C such that k >_ 1 and r''(x) = x.The minimal such k is the length of the T-cycle .5.8 .Lemma.The set B+ fl C is the union of at most finitely many Tcycles .Furthermore, B+ meez<s at mosi finitely many componente of T \ C and the points of C bordering these componente fall finto finitely many T -cycles.
Proof.This was essentially proven in [C-C 1].Indeed, by [C-C 1, (6.8), (6.18), and (6.19)] the set B+ fl C and the set of componente of T \ C that meet B+ are both finite.Let x E !3+ and let V be a neighborhood of x in R. For each integer N > 0, let WN be the set of words of length N in positive powers of {h1, . . .,h,n}_By [C-C 1, (6.5)], we see that Uf g(V n D(g)) 1 g E WN} meets B+, for all choices of V and N .From these facts the assertions follow easily .
Definition .The set 9 consists of all g E P such that (a) g maps [b 1 , bo[ diffeolnorphically onto itself; (b) giC has fixed points p < q with g'(p) = 1 = g'(q) and such that C fl ]p, q[ 7É 0 ; (c) if x E C fl ]p, q[ and if g(x) = x, then g(x) 7É 1 .5 .9 .Lemma.There are constante B > 0 and 0 < ro < 1 such that, if g E 9 and p, q are as in Me abone definition, there is a choice of h E {g, g-1 } and of x E ]p, q[ for which one of the following holds .
(3) x < h(x) < q and (q -x)1(h(x) -x) < B. The points mi and Mi lie on semiproper I's-orbits, hence there are contractions -yi and ili E rs to mi snd Mi, respectively, defined on the nonproper sides of these orbits and generating the infinite cyclic, relative holonomy groups ~-l(Fs(mi),C) and i-l(Fs(1VIi),C) [C-C 1, Theorem 1].
Let 0 < ro < 1 be such tha,t, for each value of i = 1, . . ., m, if -yi(mi) and/or r71(Mi) are not equal to 1, then they are less than ro .If all of these derivatives are 1, then 0 < ro < 1 can be fixed arbitrarily.
Clearly nao and Mi both lie in [p., cl .]C D(w) .Let m = w(mj) E [p, q] and M = w(Mi) E [p,q] .Since p. E Ii a.nd q.E Ij and Ii n h = 0, it is also clear that and we set n = w(nj) E [p, q], N = 20(Ni) E [p, q]-(4) Suppose that either 7n or M is a fixed point of g.We will show that property (1) in the statement of the lemma follows .
For definiteness, supl5ose that g(Al) = M, the argument in the alternative case being completely pa.rallel.There are two cases .
Case 1. Suppose that p. = 1Vli, hence p = M. Since C n ]p, q[ 5A 0 and since Mi is the left endpoint of a, ga.p of C, so is M = p.We take x E ]p, q[ to be the other end of the gap.Then the fa.ct that w-1(p) = p* = Mi implies that w-1 (x) = mk, some k E {1, . ..,lzz} .Also, g(M) = M implies that g(x) = x, while x E C n ]p, q[ then implies that g'(x) :~1 .Elther g or g -1 will be a contraction to x and we choose lz = gfl to be that contraction.Finally, let h* = w-1 o h o w = gtl .Then 1 > h'(x) = h*(mk), so h* is a positive power of yA and step (1) .impliesthat h'(x) < ro, as desired .
Case ,2.Suppose that p* < Mi .Then, if we take x = w(Alli) = M E ]p, q[, we can argue exactly as in Case 1 .
(5) By step (4), we can assume that neither liz nor M is fiaed by g.We will show that either property (2) or (3) in the statement of the lemma follows .We consider there possible cases .
Case 1. Suppose that w E )/Vi .Again choosing h = gtl appropriately, we see that h* (Ni) < Mi and h(N) < M < N .From step (3), ÑI and Ñ E [p, q], w(Mi) = M, and w(Ni) = N .By there remarlas and the mean value theorem, Thus, x = N satisfies property (2) in the statement of the lemina, Case 2. Suppose that w E Wj .An argument completely pa.rallel to that in Case 1 shows that ñ < h(ñ) < q and that (q -ñ)/(h(ñ) -ñ) < B, so we take x = ñ, obtaining property (3) in the statement of tlie lemina-Case 3. Suppose that -w q Wi and w q Wj .This cannot actua .llyhappen, as we now show.
We have lwi > p and w,,(Ii) n f;+ 7~0 7É w,,(Ij ) n 6+ .It follows from (5.8) that there are points yi E Ii a.nd yj E I;, belonging to r-cycles of respective lengths A(i) < A and A(j) < A, where A is as in step (2), and lo,,(yi) and w,,(yl) belong to there same r-cycles .Let k = A(i)A(j) < A' = p.Let gi and gi be the words of lengths A(i) and A(j) fixing yi a.nd yi respectively.It follows that wk=9 1 =gi(i)=h¡,o . ..oh¡, .
But then yi = wti(yi) E Ii,, so ik = i, while, similarly, yi = wk(yj) E Ii, and ik = j.This contradicts the fact that i qÉ j .
Choose r E ]ro, 1[, sufficiently close to 1 that (e6EB)-1 > 1-r .Here, of course, ro and B are as in (5 .9) .By (5.4), choose k >_ 0 so large that g(x) > r, for each g E GÓ, 0 <_ x _< bk .We want to show that every g E Gb fixes every x E ]r' (1 [0, bk] .If not, there exists g E G, and n >_ k so that g moves some points in 1r' n [bn+i, bn] C [0, bk] .Fix a choice of such g and deduce a contradiction as follows. Let g o = f -n o g o fn E I and note that this carries [b l , bol dif%omorphically onto itself, fixing b l and ao = a .Since the holonomy relative to X of semiproper leaves in X is compactly supported (4.1), we can assume that k is so large that gó(b1) = 1 = gó(a o ) .Thus, the closed set Co of go -fixed points y E X n [bl , ao] with gó(y) = 1 is nonempty.Since g o moves some points of X fl [bi,ao], there is a connected component ]p, q[ of [b l , ao] \ C o such that X f1 ]p, q[ 7É 0. This proves that go E 9 and we can apply (5.9) .
If property (1) of (5 .9)holds, then, for a suitable choice of an element h = g ±l E Gq and a corresponding choice of h o = go l , we obtain the cóntradiction that h'(f n (x)) = hó(x) < r o < r, for some f n (x) E f n ]p, q[ C [0, bk] .Thus, either property (2) or property (3) of (5.9) holds .We consider the first case, the second being entirely similar.Let x E ]p, q[ be such that inequalities p < y = ho(x) < x and (x -p)/(x -y) < B hold.Set The proof of Theorem 3 is complete .

. Geoinetry of leaves
Let M be a closed, orientable 3-manifold, F a transversely orientable CZ foliation of M by surfaces .6.1.Conjecture .If X C M is a Markov LMS of F, and if X contains no toral leaf, then there is a Riemannian metric on M relative ío which every leaf of íFIX has constant eurvature -1 .Indeed, refinements of the methods of this paper, together with standard facts about Teiclimüller space, should yield a proof of (6.1) for the case of (a) Ilyxll, for each x E X fl R; (b) IIhj il,1 < j < ni ; (c) 11wa¡ilis, for each x E C a.nd for all a, 0 E B.
Proof. .-(1) Let mi = min(C n Ii) and Nli = max(C n Ii), 1 <_ i _< m.These points border gaps of C, so we fix ni < mi and Ni > Mi in these gaps such that ni and Ni E R(h i ) .Let Ni -n2i Mi -n i B=maxl<t<~r~Ni-Mi,rni-niJ, .
of G+, we can choose the points ni and Ni of step (1) and a constant B > 0 such that w'(u)/w'(v) _< B, for each w E Wi and for all u, v E[ni, Ni]  n D(w), 1 < i < ni .