SYMBOLIC DYNAMICS FOR GEODESIC FLOWS
BY
C A R O L I N E S E R I E S University of Warwick, Coventry, England
Introduction
B y the classical result of Hopf [12], the geodesic flow on a surface of constant negative curvature and finite area is ergodic. I n the case of a compact surface the flow has sub- sequently been shown to be Anosov [2], K [17], and Bernoulli [15]. B y the work of Bowen and Ruelle [5] a n y Anosov flow on a compact manifold can be represented as a special flow over a Markov shift of finite type, with a tI61der continuous height function. R a t h e r [16] showed t h a t a n y such special flow which is K is also Bernoulli.
I n this paper we make an explicit geometrical construction of a symbolic dynamics for the geodesic flow on a surface of constant negative curvature and finite area. The construction involves the geometry of the surface and the structure of its fundamental group. The geodesic flow is shown to be a quotient of a special flow over a Markov shift, b y a continuous map which is o n e - - o n e except on a set of the first category. For a compact surface the height function is HSldcr.
The states for the Markov shift are generators of the fundamental group F, and the admissible sequences are determined b y the relations among the generators. If we lift the surface to its universal covering space the unit disc D, then admissible sequences correspond to geodesics in D which pass close to a fixed central fundamental region for F, in a sense made precise in w 3. The height function h corresponds to the time a geodesic takes to cross R, with a suitable convention if the geodesic is close to R but does n o t cut R.
The idea of our construction comes from three different sources. I n [3] Artin obtained a representation of geodesics in the Poincard upper half plane H (these geodesics are of course semi-circles centred on and orthogonal to the real axis) as doubly infinite sequences of positive integers, by juxtaposing the continued fraction expansions of their endpoints;
two geodesics are then conjugate under the action of GL (2, Z) on H if and only if the corresponding sequences are shift equivalent.
The second source is Hedlund's paper [11]. I n [14] Nielsen gave a symbolic reprcsenta-
104 C. SER~ES
tion of points on S 1 as semi-infinite sequences of generators of the fundamental group 1~1 for a surface whose fundamental region R 1 is a symmetrical 4g-sided polygon; in [11] Hed- lund represented geodesics in D b y juxtaposing the Nielsen expansions of their endpoints, showed geodesics are conjugate under F 1 if and only if the corresponding sequences are shift equivalent, and used this to prove ergodicity of the geodesic flow on D/F 1. I n [10]
he showed t h a t Artin's coding could be used to obtain similar results for H / S L (2, Z).
Finally in [13] Morse coded geodesics y in D as sequences of generators in F1 b y an entirely different method: he observed t h a t to each side of the net 741 of images of sides of /~1 under F 1 is associated a unique generator of F1, and assigned to y the sequence of generators which label the successive sides of Tll crossed b y ?. I n order to obtain a one-one correspondence between sequences with certain well-defined admissibility rules and geo- desics this coding needs to be slightly modified when 7 passes too near to a vertex of ~1 and this point occupies a large p a r t of [13]. The admissibility rules which are obtained are more or less identical with those of Hedlund.
I n view of these results, and the facts about representing a general Anosov flow as a special flow over a Markov shift, it is natural to ask whether the ideas of Morse and Hed- lund can be combined to give a representation of the geodesic flow as a special flow over some Markov shift whose symbols are generators of I ~ and where the height function measures the time to cross the fundamental region R. This is precisely w h a t we have done in this paper. Adler and Flatto (private communication) have obtained similar results in the SL (2, Z) and F1 cases above.
The symbolic dynamics we use derives from the results of [6], in which the action of t h e f u n d a m e n t a l group on S 1 is shown to be orbit equivalent to a certain Markov m a p / r of finite t y p e acting on $1; t h a t is, x =gy, x, y E S 1, g E F ~ / ~ ( x ) =/~(y) for some n, m~>0.
We copy Artin and Hcdlund in representing geodesics in D b y juxtaposing the/-expansions of their endpoints, and t h e n show t h a t these sequences have a geometrical interpretation analogous to Morse's idea of listing successive crossings of the fundamental region R.
Finally we derive the representation of the geodesic flow on D/F as a quotient of a special flow over the natural extension o f / r .
To understand the constructions the reader will need to be familiar with the m a p s / r of [6]. I n [6] we first c o n s t r u c t e d / r for groups F whose f u n d a m e n t a l region R could be chosen to satisfy a certain s y m m e t r y condition (*), and then showed t h a t a n y F could be deformed b y a quasi-conformal deformation to a group F ' satisfying (*). We then carried over the definition o f / r " using the b o u n d a r y homcomorphism and constructed the general /r. We shall a d o p t the same procedure here, so t h a t in the main p a r t of the work, w 1-w 4, we shall only be concerned with groups whose fundamental region satisfies (*).
S Y M B O L I C D Y I ~ A M I C S F O R G E O D E S I C F L O W S 105 I n w 1 we review briefly the definition and properties o f / r and t h e n determine which sequences of generators correspond to admissible/-expansions. I n w 2 we describe the F action on S i in terms of sequences and show how to juxtapose sequences to represent certain pairs of points on S 1. I n fact geodesics are conjugate under F if and only if the corresponding sequences are shift equivalent.
I n w 3 we discuss the relation of this representation to the listing of successive crossings of R and in w 4 derive the symbolic representation of the flow. Finally in w 5 we show how to carry these results over to the general case using quasi-conformal maps.
We shall keep to the notation of [6]. I n particular, when describing arcs on S i, we always label in an anti-clockwise direction, so t h a t
PQ
means the points lying between P and Q moving anti-clockwise from P to Q. We write(PQ),
[PQ], etc., to distinguish open and closed arcs on S i.Throughout, I" is a finitely generated Yuchsian group of the first kind acting in the unit disc D; t h a t is, a discrete group of linear fractional transformations
z~-~-(az +b)/(cz
+d),ad-bc = 1,
which m a p D to itself and such t h a t there are points on S ~ with dense orbits.The corresponding surface
D/D
is a R i e m a n n surface of constant negative curvature and finite area; we are concerned with the geodesic flow on the unit tangent bundleM of D/F.
l" has a fundamental region R in D which can be t a k e n to be a polygon bounded b y a finite n u m b e r of circular arcs orthogonal to S i. A vertex of R lying on S i is called a cusp.
D / F is compact if and only if R has no cusps. Geodesics on
D/F
are the projections of circular arcs in D orthogonal to S i.I f gel",
g(z)=(az+b)/(cz+d),
then the circle]cz+d[
= 1 is called the isometric circleof g,
because [g'(z)[ > 1 inside this circle and [g'(z)[ < I outside. The isometric circle is always a circle orthogonal to S i.I suspect the idea t h a t something like the ideas of this p a p e r might work has occurred to a n u m b e r of people. I n particular, see the r e m a r k at the end of [10]. Certainly it had to b o t h Adler and Moser, and I would like to t h a n k both for the benefit of useful conversations.
w 1. Symbolic representation of points on
S iL e t us recall briefly the constructions made in [6]. As explained in the introduction, F is a finitely generated Fuchsian group of the first kind acting in the unit disc D. F has a fundamental region R which consists of a polygon with a finite n u m b e r of sides (s~}~-i;
these sides extend to circular arcs
C(s~)
orthogonal to S i. Each side st of R is identified with another sideA(s~)
b y an elementgi=g(s~)ED;
the set F0--(g~}~l is a symmetrical set of generators for r . The images of the sides (s~} under F form a net ~ in D. We will say R satisfies p r o p e r t y (*) if:1 0 6 c . SERIES
(i) C(s) is t h e isometric circle of s, a n d (ii) C(s) lies c o m p l e t e l y in ~ .
Throughout w 1-w 4, we shall assume R satis/ies (*) and moreover that R is not a triangle and does not have elliptic vertices o/order 2. (See [6].)
A t y p i c a l f u n d a m e n t a l region is shown in Fig. 1. (See also Fig. 1 of [6].)
W e label t h e sides of R, s~, s~ ... s~ in anti-clockwise order; t h e v e r t e x v~ is t h e inter- section of ss-1 a n d s~ (with s o =s~). C(st) m e e t s S ~ in P~, Qs+l, so t h a t t h e order of points along C(ss) is Ps, vt, Vl+l, Qt+l-
/ = / r : S L + S i is defined b y / r ( X ) =gs(x), xE[PsPt+l). I n [6] we showed t h a t / r has t h e following properties:
(a) E x c e p t for a finite n u m b e r of pairs of points x, yES1:
x = g y , x, y E S 1, g E P ~.3n, m>~O such t h a t /=(x)=/m(y).
(b) / is M a r k o v in the following sense:
T h e r e is a finite or c o u n t a b l e p a r t i t i o n of S 1 into intervals { s}s=l such t h a t I
(Mi) / is strictly m o n o t o n i c on each Is a n d e x t e n d s t o a C 2 function [~ on _rt, (Mii)/(Ik) N I j ~ O ~/(Ik)~_ Ij, Y], It,
(Miii) ( J ~ 0 / r ( I ~ ) ~ - I k , V], It,
(Miv) I f / s = [at, bs] t h e n
(L(as), L(bs)}~
is finite.Moreover t h e p a r t i t i o n {Is} is finite if a n d only if D / F is c o m p a c t , or e q u i v a l e n t l y if R has no cusps.
(c) (Ei) I f t h e r e are no cusps, t h e n 3 N > 0 such t h a t inf [ (x)] > r >
x e S 1
(Eii) A cusp of R is a periodic point for / with d e r i v a t i v e one. There is a sub- set K ~ S 1, consisting of a union of intervals Is, so t h a t if /K(x)=/~(X)(x), n ( x ) = rain { n > 0 : / n ( x ) E K } , x E K , is t h e first r e t u r n m a p induced on K , t h e n 3 N such t h a t infxEK I > r > 1.
T o each p o i n t x E S 1 we can associate a s o - c a l l e d / - e x p a n s i o n (cf. [1]). T h e usual w a y to do this is to write X = i o i l i 2 ... if In(x)EI~, n = 0 , l, 2 . . . (There is a slight a m b i g u i t y a t t h e endpoints which we shall clarify below.) B y (Mii) t h e rule d e t e r m i n i n g which se- quences ioili ~ ... can occur is of finite t y p e [8]; n a m e l y iri s occurs i f f / ( i r ) D i~.
F o r our purposes it is b e t t e r to label points using the generators F 0 of F, so we replace t h e p a r t i t i o n {it} b y {[PtPs+l] = [gs]}. T h e rules d e t e r m i n i n g which sequences are admis-
S Y M B O L I C D Y N A M I C S F O R G E O D E S I C F L O W S 107 sible is no longer of finite type. W e s a y a sequence e1% ... enEF~ is admissible if [.J~=l/-r([e~l])~=O. L e t ~+ =(e1% ...
er : eke +l
... e~+, is admissible Vie, l e N } . Define~: Z + - ~ S 1 b y g(ele ~ ...)-~ [') r~=l ]-r([e~l]). T h e intersection is n o n - e m p t y since this is t r u e of all finite intersections a n d it contains at m o s t one point because of t h e e x p a n d i n g condi- tion (c). W e discuss t h e t o p o l o g y of Z + a n d c o n t i n u i t y of ~ in w 4.
To see which sequences e1% ... belong t o Z+, it is e n o u g h t o find those sequences el e2... em for which [7 ~= 1/-r((er 1)) ~ O , where (e~) = I n t [e~].
To state t h e rules we need some more terminology. Starting at a v e r t e x v~ with t h e side s~ a n d g e n e r a t o r g~, we get a cycle of vertices v~ = w 1 .. . . . wp a n d corresponding generators g~ = h 1 ... h~. ([9] Sec. 26 a n d [6] L e m m a 2.4.) W e s a y t h e anti-clockwise sequence h~ 1 h~l ... h~ 1 is in left-hand (15) cyclic order. Similarly, starting at v~+ 1 with side s~ a n d g e n e r a t o r g~
we get a cycle v~+l=Zl, z 2 .. . . . zq a n d generators g~=]l, ]2 . . . jq. W e s a y t h e clockwise sequence ]~1]~1 ... is in r i g h t - h a n d (R) cyclic order. There exist integers #, v such t h a t ( h ~ l h ~ l ... h~X)~,= (];~]~1 ... ]~1)~ = 1 . p # a n d qu represent t h e n u m b e r of sides of Tl which m e e t a t t h e vertices v~, v~+ 1 respectively, a n d therefore b y (*), p # = 2 1 , q v = 2 k are even (see Fig. 1). W e call L cycles of lengths l - 1 , l, l + 1, D-(defieient),//-(half), a n d S-(super- fluous) L cycles respectively, a n d similarly for R cycles of lengths k - 1 , ]c a n d ]c + 1. A cycle of length 21 or 2/c is called full. Notice t h a t a full cycle is equal to t h e i d e n t i t y in F.
I f h =g~, write h+ =g~+l a n d h-=g~_~. I f B = b I . . . b~, B 1 =b~ ... br+~, C = c 1 ... c~ are L cycles with c~ ~ = (br+~l)+, we s a y B a n d C are a d j a c e n t or consecutive L cycles; similarly if B, B ~ a n d C are R cycles a n d c ; 1 = (b~+~) - we s a y B, C are consecutive R cycles (see Fig. 2). A sequence B~ ... B~ of consecutive L cycles, where B~, B~ are H-cycles a n d B~ . . . B~_~
are D-cycles, will be called a L H-chain; such a sequence with B 1 a L D-cycle is a L D-chain.
Often we represent a chain symbolically b y D D ... D H .
I n Figs. 1 a n d 2 we indicate t h a t t h e side s~ of R is associated to g~eF 0 b y an arrow pointing i n t o / L W e write ( g ~ } for t h e interval [P~P~+I) (the inverse is to m a k e subsequent c o m p u t a t i o n s work properly) a n d write x = g j ~ ... i f / ~ - ~ ( x ) ~ (g~,}, n = 1, 2 . . .
P ~ O P O S I T I O ~ 1.1. A sequence e 1 ... %, e~EF0, is admissible i / a n d only i/
(1) gg-t, g El~0, does not occur.
(2) N o R / / - c y c l e s occur.
(3) N o L S-cycles occur.
(4) N o L H - c h a i n s occur.
Proo/. Referring to Fig. 1, let P ~ = C k , P i + I = C 1 , Q~=D1, Q~+I=DI. T h e arcs zlC1, ZlC 2 .. . . . z l C ~ are t h e arcs of t h e n e t ~ e m a n a t i n g f r o m z 1 a n d lying within the isometric
108 C. S E R I E S
D~.
Di
= Q~ C k = H 1 = P~K
} / /v
s~
+ IJ~ = gF)l \\
_ _ R \ :Bt
\
\
h2
i,~ \\ j~ _
/ A . /
A s A 1 Ao Fig. 1
circle C(8~) of g~; s i m i l a r l y t h e a r c s
w l D 1 ... w l D l
a r e t h e a r c s of ~ e m a n a t i n g f r o m w 1 a n d l y i n g w i t h i nC(s~).
B y [6] L e m m a 2.2,wlDz_ 1
a n dzlCk_ 1
d o n o t i n t e r s e c t , wl, w~ . . . w v is t h e v e r t e x cycle s t a r t i n g a t w 1 w i t h side s~ a n d hi, h 2 .. . . . h v is t h e c o r r e s p o n d i n g cycle of g e n e r a t o r s . S i m i l a r l y z 1, z 2 .. . . . zq is t h e v e r t e x cycle s t a r t i n g a t z 1 w i t h side s~, w i t h corre- s p o n d i n g g e n e r a t o r s ix, ?'2 .. . . . ?'q.wlH1 ... wlH~; zlL1 ... ZlLk; z~Ao, z~A1 ... z~Ak;
a n dw2Bo, w 2 B 1 ... w~B z
a r e all t h e a r c s of T / l y i n g inside t h e i s o m e t r i c circles ofh~ 1, ?'~1, ?'3
a n d h 2 r e s p e c t i v e l y . G, F a n d K a r e t h e e n d p o i n t s ofC(h~), C(j~), C((h~l) -)
l y i n g insideC(h2),
C(?'2),C(h~ 1)
r e s p e c t i v e l y a n d J is t h e e n d p o i n t of t h e arc of ~ / t h r o u g h v~_ 1 a d j a c e n t t o b u t o u t s i d eC(h;I).
SYMBOLIC DYI~AMICS FOR GEODESIC FLOWS
109c~ 1 = (bV+ll) +
Fig. 2(~a). Consecutive L e y e l e s ~ /
(b.~+ll)_ = cll / //<b~-l>
Fig. 2(b). Consecutive R cycles
(At a parabolic vertex, 1 = ~ a n d we label points as H ~ , H ~ _ 1, Ho~_2 .... etc. a n d in c o m p u t a t i o n s t r e a t ~ e x a c t l y as a n y o t h e r integer.)
Notice t h a t t h e m a p g~ carries D~, z 1, Wl, Ck o n t o A 1, z2, w~, B x respectively; C 1 ... Ck-1 onto A s ... Ak a n d D 1 ... Dz-1 o n t o B 2 ... B~.
N o w
]]Evk.v,)=hl=jl.
]([CkC1) ) covers all intervals <h> except <j~l>, <hi> a n d<h~l>. Since ](<hfl>) f~ <hi> = O , we get (1). /([CkC~)) f3 <iffl> =[AkA~+I ), 1
<~r<~k-2
a n d[([CkCk-x)) f?
<]fix> = O .Moreover]([CkC~) ) f3
<h>=/([CkC1) ) f?
<h> for 1<<.r<~k-1
a n dh 4 ] ~ 1.
Therefore the sequence
]fx]~x ... ]~1
is n o t admissible, b u t otherwise t h e restrictions fol- lowing t h e symbols ]fx ... ?.~1 r ~< k - 1, are t h e same as those following ]71 alone. R u l e (2) above follows.Similarly we h a v e
/([GkG1) ) f3 <h~-l> = [ B I G ) ,
/([ DrC1) ) N
<h~l> =[ Br+I G),
/([D~-1~1)) n <h;~> =l <~ r <~ l - 2,
110 C. SERIES and
a n d
I([D~Ct)) tl <h> = [([CkCt)) N <h> for 1 ~< r ~< l - 2 , h # h ~ ~, [([D~_tCt) ) fl <h> = [([CkCt) ) N <h> for h # h ~ ~, (h+) -1
l([Dz_~ Ca)) fl <(h~)-~> = <(h;)-~> - [ G B z ) . Therefore t h e sequence h f l h ~ 1 ... h[-+11 is n o t admissible, which is rule (3).
T h e only restrictions following h~ 1 ... h j 1, r < l , are t h e same as those following h~ -1 alone. Following hs 1 ... h[-lh, where h=~hi-11, are t h e same restrictions as after h alone.
- - 1 ~ - - 1
After h f I ... hi (hi+l) is t h e same restriction as after k - l ( h + l ) -1, where k -1 is t h e element preceding (h{+l) -1 in t h e L order. T h u s (h~+x) -t is n o t t h e first element in a L H-cycle; also if (h~t) - t is t h e first element of a L D-cycle which ends in s -t, followed b y (t+)-x where s - t t -1 are in L order, t h e n (t+)-t is n o t t h e first element of a L H-cycle.
R e p e t i t i o n of this a r g u m e n t gives rule (4), a n d we h a v e examined all t h e possibilities for finite sequences e t ... %. Y~+ therefore consists of all sequences ele 2 ... in which each finite block satisfies (1)-(4) above.
T h e m a p ~: E +-+S t is of course n o t bijeetive. More precisely x E S t has two representa- tions in Y,+ whenever ]k(x)E{P~}~=t for some k~>0. P~ can be written either as D D D ....
a n infinite string of consecutive R D-cycles, or as H D D .... an infinite string of consecutive L cycles.
Convention. I n order to keep t r a c k of w h a t is h a p p e n i n g we shall in f u t u r e a d o p t t h e following rule:
W h e n e v e r x E S t has two symbolic expressions i n Y~+, we write x = e t e 2 ... where ere 2 ...
is the expression [or x ending i n L cycles.
This is equivalent to a t t a c h i n g P~ to t h e interval (P~P~+t) r a t h e r t h a n (P~_IP~).
Also notice ~ a ( e ) = ] z ( e ) , e E E +, provided e does n o t end in an infinite string of R D-cycles, where a is t h e left shift on ~+.
R e m a r k 1.2. I n t h e ease where R is a s y m m e t r i c 4g-sided polygon, our rules are identical w i t h those of [13] p. 77 a n d closely related to those in [11] p. 791.
w 2. Representation of geodesics in D
W e would n o w like to represent a geodesic y in D b y t a k i n g t h e / - e x p a n s i o n s of its endpoints P , Q, s a y P = e l e 2 .... Q = / 1 / 2 ... a n d writing y . . . . /2/lele2 .... U n f o r t u n a t e l y , t h e sequence so obtained m a y n o t be admissible according to t h e rules of w 1. There are
SYMBOLIC DYNAMICS FOR GEODESIC FLOWS 111 two problems: (i) Is the reversed sequence .../~/1 always admissible? And if so: (fi) W h e n is . . . / J l e l e ~ ... admissible? The answer to (i) is no. I t is perhaps more natural to consider the inverse sequence .../~1/~1. This is however still in general inadmissible. To circumvent this difficulty we use the following trick:
[ - e x p a n s i o n s . RecM1 t h a t in defining / we m a d e an a r b i t r a r y choice t h a t / I ceded+l)=g~.
We could equally well have taken/I(Q~ 1Q~]=g~; let us call this m a p [. [ obviously has exactly the same properties a s / , and the admissibility rules are obtained b y interchanging
' R ' and ' L ' in Proposition 1.1 above.
L E M M A 2.1. L e t e 1 e 2 ... be a n a d m i s s i b l e s e q u e n c e / o r / . T h e n the i n v e r s e s e q u e n c e . . , e ~ l e ~ l i s a d m i s s i b l e / o r ], a n d vice v e r s a .
P r o o / . This follows easily from the remarks above, since an R cycle in ele ~ ... becomes
an L cycle in ... e~le~l; a n d consecutive R cycles become consecutive L cycles.
L e t P, Q E S 1 and let P = e le 2 . . . . Q =/1/3.-. be t h e / - and/-expansions of P, Q respectively.
We shall call the directed geodesic ? joining Q to P admissible if Q-1. p . . . . / ~ 1 / ~ l e l e 2 ...
is admissible, and we shall also write ? = . . . / ~ l / 1 - 1 e l e 2 . . . . Below in w 3 we shall see t h a t admissible geodesics pass 'close' in a certain sense to the fundamental region R. This will deal with problem (ii) above.
L e t Z be the space of doubly infinite admissible sequences (i.e. all finite blocks satisfy- ing (1)-(4) of Proposition 1.1) with left shift m a p a.
To proceed we need to know something a b o u t the action of I~0 (the set of generators of F) on S 1 in terms of the symbolic representation of w 1.
P R O r O S I T I O N 2.2. L e t x = e l e 2 ... EZ +, g E F 0. T h e n
(1) g ( x ) = g e l e 2 ... w h e n e v e r g e l e 2 ... e Z + a n d
(2) g ( x ) = e ~ e a ... i / g = e ; I.
P r o o / . We refer again to Fig. 1 with g = h 1.
(1) Suppose g e l e 2 . . . is admissible. Then
(a) g e l e ~ ... does not begin with a R H-cycle.
(b) g e l e 2 ... does not begin with a L H-chain.
(C)
el:~g -1.
Observe t h a t g e l e 2 ... begins with a R H-cycle iff x = e l e 2 . . . E [ H 2 H 1 ) ; g e l e 2 ... begins with a L H-chain iff x E [C 1 D1). Therefore (a), (b), (c) together imply x ~ [H 2 Dz).
Since x ~ C(g), the isometric circle o f g, g ( x ) E C ( g -1) N $ 1 = ( g ) U [BoBI) (cf. [9] Sec. 11).
112 C. SERIES
But
g(x) ~
[B 0 B1) since X ~ [H 2/tl). Therefore g(x) E (g~, so/(g(x)) = g - l ( g ( x ) ) = x = e 1 e~ ... and g(x) = g e l e 2 ....(2) Suppose g = e ~ 1. Then x E ( g -1) and so/(x) = g ( x ) and g ( x ) = e 2 e a ....
I t is possible to derive rules for the action of F 0 on Z + in general. As this is not neces- sary for our development and the details become rather involved, we state without proof:
P R O P O S I T I O ~ 2.3. S u p p o s e x E S 1, a n d g E F . Let x = e l e ~ .... g(x) =/1/2 ... be t h e / - e x p a n - sions o / x , g(x). T h e n 3s, t > 0 so that gele 2 ... et=/1/~ .../8 in F a n d et+r = / s + , r > 0 .
Of course we have already proved the second p a r t of this s t a t e m e n t in [6], see p r o p e r t y (a) o f / r in w 1.
This proposition is of interest because it enables us to prove the analogue of the results of Hedlund and Artin mentioned in the Introduction, t h a t admissible geodesics are con- jugate under F iff the corresponding sequences are shift equivalent. The proof is an easy consequence of Proposition 2.3:
P R O P O S I T I O ~ 2.4. Let ( P , Q ) , (R, S ) E S l x S 1 be such that Q-1. p , R-1. SEy~. T h e n 3 g E F with g P = R , g Q = S i// 3 n E S with a'(Q-~.P)=S-I. R .
Proo/. L e t P = e l e 2 .... Q = / 1 / 2 ... be t h e / - and ]-expansions of P, Q respectively. We have . . . / ~ l / ; l e l e ~ ... EZ. B y Proposition 2.2,
e~l(P) = e2e a ... and e;l(Q) =e;1/1/~ ....
Hence (~(Q-I.p) = (e~lQ)-l. ( e ~ l p ) .
Conversely, suppose P, Q E S 1 and g E F are such t h a t Q-1. p , (gQ)-l. ( g p ) E Z . B y Proposi- tion 2.3, we have
P = e~ . . . e n e n + 1 . . . and g P = U 1 . . . U m e n + 1 . . .
where g e I ... e n = u 1 . . . u m .
Similarly, Q =/1 ---/~/~+1 .... gQ = v l ... vq/~+i ... and g/x . . . / ~ = v I ... vq.
Thus u I ... Umen 1 ... e; 1 = v 1 . . . V q / p 1 ... /~1 and so
Q-~.P . . . . / p : l / p 1 . . . / l i e I ... e n e n + 1 . . . a n d (gQ)-~. (gP) . . . . / p l l v q l ... v l l U l ... U m e n + 1 . . .
are shift conjugate.
This result is sufficient to show t h a t the geodesic flow on D / F is ergodic, b y the method used b y Hedlund in [11]. ]Notice t h a t the restriction to admissible geodesics with Q-.1P E E corresponds to the restriction in [3] t h a t t h e endpoints of geodesics lie in ( - 1, 0) a n d (0, oo). F o r a discussion of the relevant measures, see R e m a r k 4.4 below.
S Y M B O L I C D Y N A M I C S F O R G E O D E S I C F L O W S 113 W e shall instead follow t h e m e t h o d of Morse to o b t a i n a representation of t h e geodesic flow itself.
w 3 Crossings of the fundamental region R
W e n o w w a n t to investigate in detail t h e relationship between t h e symbolic expansion 7 . . . . ] ~ l / s 2 ... of an admissible geodesic a n d t h e order in which 7 cuts successive sides of t h e n e t 7/. Recall t h a t each side of R is labelled b y a u n i q u e element g E F 0. This label can be t r a n s l a t e d b y a n element of F t o assign a u n i q u e element of F 0 to each (oriented) side of ~ . T h e idea t h a t 7 should cut successively sides .... ]~1,/s el ' e~ .... m a y unfor- t u n a t e l y fail when 7 passes too close t o vertices in ~/. W h a t we shall show is
T ~ E O R E M 3.1. F o r a n y e E Z , w i t h corresponding directed geodesic 7, there is a dis- tinguished copy R ( 7 ) o] R such that
(1) 7 t3 R ( 7 ) # f D
(2) r n ~ # o ~ R ( ~ ) = R
(3) 7 cuts i n succession R ( 7 ) , ( r - l R ( a ? ) , ... where (r -n = e 1 ... en /or e . . . . / ~ l / ~ l e l e 2 ....
T h r o u g h o u t this section, b y R we shall m e a n t h e open region b o u n d e d b y t h e sides st.
S t a t e m e n t (3) needs a little i n t e r p r e t a t i o n w h e n 7 is a geodesic which goes t h r o u g h a v e r t e x v of 7/. L e t R 1 ... R2k be t h e copies of R meeting a t v, in anti-clockwise order r o u n d v. I f 7 passes f r o m R 1 to Rk+l we s a y 7 cuts R 1, R~k ... Rk+l in order. I f 7 coincides with t h e side of ~ between R 1 a n d R2, we s a y 7 cuts R1, R2k ... Rk+2 in order a n d if ? coincides with t h e side between R 1 a n d R~k, 7 cuts R2k .... , R~+I.
The idea of T h e o r e m 3.1 is t h a t if 7 fi R = O , 7 can be deformed b y a sequence of 'small deformations' to a curve ~ such t h a t ~ f3 R~=O which cuts R, a - l R in order. This sequence of deformations will determine R(?).
L e t us m a k e this m o r e precise. As above, let v be a v e r t e x of 7 / w h e r e copies R 1 ... R2~
o f / ~ meet, in anti-clockwise order r o u n d v. L e t Wr, 1 ~<r~<2k, be t h e v e r t e x of ~ a d j a c e n t to v, along the side between Rr a n d Rr+ 1 (see l~ig. 3), a n d let A~ be t h e e n d p o i n t of this side on S 1.
A directed curve fl will be said to pass n e a r v ff it passes f r o m R 1 t o Rk+l c u t t i n g t h e arcs [vw~), 1 ~< r ~< k, or [vwr), 2k >~ r ~> k + 1, in order, see Fig. 3. I f fl cuts [vw~), 1 < r <<. k, let/~ be a curve which coincides with fl everywhere except near v, where it cuts instead t h e arcs (vwT), 2k ~> r/> k + 1. ~ is ' a small d e f o r m a t i o n of fl r o u n d v'. R2k-r+2, 2 ~< r ~< k, is called t h e c o n j u g a t e region t o R , R2k_~+2=R *(z''). I f fl cuts [vwr), 2 k > ~ r > ~ k + l , we write R ~ = R *(~''~, 2 k > ~ r > ~ k + 2 a n d call Rr self-conjugate. W e write *(fl, v ) = * where there is no a m b i g u i t y .
8 - 8 0 2 9 0 7 Acta mathematica 146. I m p r i m 6 1r 4 M a i 1981
114 C. S E R I E S
Ak+l
.R2z ~ I'
Rk+l
~ A g w~
g~
7/
wl ~ A 2
i
A 1
Fig. 3
We shah call a curve obtained from fl b y a sequence of small deformations a deforma- tion of ft. We m a k e the same conventions a b o u t the order of regions cut b y a deformed curve ~ through a vertex, as for geodesics y.
Notice t h a t the conjugate of a region S is a locally constant function of ~.
LEMMA 3.2. I / the/undamental region R constructed in [6] w 3 has/our sides, then at least eight sides meet at a vertex.
Proo/. I t is straightforward to check all the cases in [6] to verify t h a t R always has more t h a n four sides, unless the signature of F is {1; 1; vl}. B u t since v1~>2, and the corre- sponding R has interior angle ~/2Vl, we see t h a t in this case a t least eight sides m e e t a t a vertex.
COROLLARY 3.3. There are no triangles/ormed by ~ . I / / o r edges o/ ~ /orm a quadri- lateral, then at least eight sides meet at a vertex.
Proo/. Suppose the triangle or quadrilateral is not already a fundamental region. Then there is a vertex v of T / o n the interior of one of the sides of t h e region. Any other edge of
S Y M B O L I C D Y N A M I C S F O R G E O D E S I C F L 0 W S 115 through v forms a smaller triangle or quadrilateral. Proceeding in this way we eventually reach a region of minimal size which must be a copy of R.
Lw~MA 3.4. I n a sequence o / s m a l l deformations o/ a geodesic ~, a region S is associated to at most one conjugate region S*, across a unique vertex v. Likewise S* is the conjugate of at most one region S.
Proof. If s is a side of S, let B ( s ) E S 1 be the arc of S 1 interior to the circle C(s). Notice t h a t if ~ is obtained from y b y a sequence of small deformations, and if S * ~ S is obtained b y a deformation of ~ across the vertex v of S, and if s, s' are the sides of S meeting at v, then ~ has one endpoint in B(s) - B(s') and the other in B(s') - B(s).
Similarly, if ~ is a deformation of ? across a vertex w, at which meet sides t, t' of S, with conjugate region S*'-~S, then ? has its endpoints in B ( t ) - B(t'), B ( t ' ) - B(t).
If u, u' are sides of S then since extensions of non-adjacent sides of S do not meet ([6] L e m m a 2.2), we have B(s) n B(t) = O unless s =t or s, t are adjacent. After interchanging s w i t h s' and t with t' if necessary, there are three cases:
Case 1. s = t , s' = t ' . Then v = w and clearly S* =S*'.
Case 2. s = t , s ' ~ t ' . B ( t ' ) - B ( t ) is disjoint from B ( s ) - B ( s ' ) , so B(t') n B ( s ' ) ~ Q since it contains an endpoint of y. Then t', s' are adjacent. B u t this means R has only three sides, s, t', s', which is impossible.
Case 3. s=4=t, s'@t'. Without loss of generality, we m a y suppose ( B ( t ) - B ( t ' ) ) N ( B ( s ) - J B ( s ' ) ) ~ O . Then s, t are adjacent. I n this case we also have B(t') n B(s')~=O, since this set contains an endpoint of y. Hence s', t' are adjacent. Then R has four sides, s, s', t and t'. Since non-adjacent sides of R do not meet, y has its endpoints in sectors of the vertex star at v separated b y one sector only, namely t h a t containing S. B u t since by L e m m a 3.2 at least eight copies of R meet at v, the endpoints of y do not then lie in dia- metrically opposite sectors at v. Then y does not pass near v, which is contrary to assump- tion.
The final statement is proved b y exactly the same argument.
Thus we m a y write S * = S*(V), independent of v and the deformation ~.
L~MMA 3.5. Let ~ be a geodesic. Then ~ cuts a region S at moat once, and i / y N S ~ and S=~S*, then 7 n S* = 0 .
Proof. If y cut S more t h a n once, then =~ (~ n ~S) > 2. B u t =~ (~ fi ~S) ~< 2, since S is geodesieally convex. (This uses the fact t h a t the interior angles of S are all less t h a n •, and the formula A = 7 ~ ( n - 2 ) - ~ ~ for the area of a geodesic polygon.)
116 c. s ~ a ~ s
S u p p o s e 7 p a s s e s n e a r t h e v e r t e x v of 2 a n d sides s, s' m e e t a t v. H ~, ~ 2*~ff) t h e n 7 w o u l d h a v e t o cross t h e e x t e n s i o n s C(s), C(s') of s, s' twice, w h i c h is i m p o s s i b l e .
L ~ x 3.6. Let ~ be a de/ormation o~ a geodesic 7. Suppose 7 cuts in order R1 . . . . , Ra (with the above conventions i] ~, passes through a vertex o/ ~). Then 7 cuts in order ~ ... ~ , where ~ is one o/R~, R~.
Proo/. This follows e a s i l y b y i n d u c t i o n on t h e n u m b e r of s m a l l d e f o r m a t i o n s . F o r one d e f o r m a t i o n i t is clear f r o m t h e definitions.
CO~O~,LAR:r 3.7. Let ~ be a de/ormation o / a geodesic 7 and suppose ~ (] 2 ~ . Then either 7 f] S=4=~ or there is a unique region 2~ with 7 f~ S ~ and S = 2 " .
Proo]. L e t .... R I, R~ . . . . b e t h e sequence of regions c u t b y 7" B y L e m m a 3.6, 2 = R~
R* t h e n N / ~ = ~ . S u p p o s e or R* for s o m e i. I f 2 = R ~ we a r e done. I f 2 = R * a n d R ~ = ~ 7
7 ~ ~ q ~ a n d t h e r e is a r e g i o n T ~ R ~ w i t h 7 N T ~ , T* = 2 . T h e n T = R ~ for s o m e }" a n d R* = R*. B y L e m m a 3.4, R~ = R~.
L w ~ M A 3.8. Let v, It x .... , 1 ~ be as in Fig. 3. Let ~ be a geodesic with endpoints in (A2kA1) , (AkAk+l) , cutting in order R~, R a ... R~. Then there is a de/ormation 5 o / ~ which cuts in order ~1, R2k, ..., Rk+x.
Proo]. L e t xo=v , x l = w l , x 2 .. . . ; Y0 = v , yl=wk, Y2 .. . . be t h e v e r t i c e s of ~ a l o n g [vA1), [yAk) a n d s u p p o s e a c u t s [vA1) on [x~x~+l) a n d [vAk) on [YqYq+l). L e t l be a n y edge of ~/
t h r o u g h u E (x~}~, o t h e r t h a n AxvAk+ 1 o r AkvA2k. 1 h a s a n e n d p o i n t L in (AIAk) , o t h e r w i s e l, AlvAk+ 1 a n d AkvA~k w o u l d f o r m a t r i a n g l e . L e t z be t h e v e r t e x of ~ a d j a c e n t t o u o n [uL). L e t m be a side of ~ / d i s t i n c t f r o m l t h r o u g h z. W e can s u p p o s e m h a s one e n d p o i n t in (LAk), for o t h e r w i s e l, m, AkvA2k a n d AxvAk+ 1 f o r m a q u a d r i l a t e r a l . I n t h i s case p i c k mX4=m, [ t h r o u g h z (possible since >/8 sides m e e t a t z). T h e n e i t h e r m x, m, vAk f o r m a t r i a n g l e , w h i c h is i m p o s s i b l e , o r m 1 h a s a n e n d p o i n t in (LAk). T h e o t h e r e n d p o i n t of m 1 lies in (AxL), o t h e r w i s e m 1, l a n d vA 1 f o r m a t r i a n g l e .
T h e n e i t h e r a ~ l E [uz), o r m 1 c u t s a t w i c e o r t o u c h e s ~, b o t h of w h i c h a r e i m p o s s i b l e . So ~ n l e [ u z ) .
W e n o w see a passes n e a r x~. F o r b y t h e a b o v e , a c u t s e v e r y side of ~ t h r o u g h x~
b e t w e e n x~ a n d t h e a d j a c e n t v e r t e x of ~ / i n t h e d i r e c t i o n of (AxA~). D e f o r m i n g r o u n d xz, we see r e p e a t i n g t h e a r g u m e n t 5 p a s s e s n e a r x~_ x, etc.
S i m i l a r l y g c a n b e d e f o r m e d r o u n d yq, yq_x . . . L e t ~ b e t h e c u r v e o b t a i n e d b y d e f o r m -
S Y M B O L I C D Y N A M I C S F O R G E O D E S I C F L O W S 117 ing successively round x~ ... xl, Yq .... , Yl. Then o2 passes near x 0 = v, and deforming round v we get the required result.
L e t W = ( p E s I : P is the endpoint of a geodesic in ~ / t h r o u g h a vertex of R}.
PROPOSITION 3.9. Suppose 7=Q-1.PEY~. Then 7' can always be deformed to a curvey*
which cuts R, 7-1R in succession, unlezs possibly P E W or Q E W. In this case either 7 is a side o/ ~l and cuts R, (r-l R in succession or 7 is not a side o/ TI and there are geodesics 7'=
Q,-1.p, EZ arbitrarily close to 7, with P', Q' ~ W.
Proof. We refer throughout to Fig. 1. W i t h o u t loss of generality we m a y assume PE[CkC1). This means a_l =g?.l g~l/~ is the copy of R adjacent to R along st.
I f Q lies outside all the circles C(8~_2) , C(s~_x) , C(s~), C(S,+l) it is clear t h a t 7 N R ~ : O , and t h a t either 7 N (s~)~=O, or 7 N (S~_i]=~=~. ((84_1] = (Vi_lVi].) I n the first case 7 cuts in succession R, a - l R . Otherwise P E [Ck D1). I f P E (Ck D1), we are in the situation of L e m m a 3.8 relative to v~, so 7 can be deformed to cut R, a - l R in order.
I f P = Ck then 7' = Q-1.p1 where p1 e (Ck D1) is admissible. I f Q e (C~ Dz] then Q-1p ~ E.
Suppose Q E (LrLr+l] 1 ~<r ~</c-1. Then the f-expansion of Q begins with an L cycle of l e n g t h / c - r . Since Q-~.PEE, P begins with an R cycle of length at most r - l , so t h a t P E [Ck Ck-r+l). This means 7 lies outside the circle Lr v~+l Ck-r+l, so 7 N/~ ~ O, and 7 cuts a - l R after R.
Suppose QE(H~+IH~] , l ~ < s ~ < l - 2 , or Qe(KHt_I] and s = l - 1 . The f-expansion of Q begins with an R cycle A r If A 1 is followed b y consecutive R cycles A2, ..., A~ of lengths D ... D, H respectively then A~ has length 1 - s - l , otherwise A 1 has length l - s . There- fore since Q-1P6F~, if P begins with a n / 5 cycle B1, and B1 is followed b y consecutive L cycles B~ ... B m of lengths D .... , D, H then B 1 has length a t m o s t s - l ; otherwise B~ has length a t m o s t s. This means t h a t P e [D~_~ C1).
Now if 7 N R ~ = ~ the result is obvious. Otherwise unless P=D~_s or Q=Hs, or 7 is a side of ~ , we are in the situation of L e m m a 3.8, with Q, P in the diametrically opposite sectors (H~+~Hs), (Dt_sD~_~+~) at v. Applying L e m m a 3.8 we get the required deformation.
I f P -- D~_~ or Q = Hs, and P ' e (D~_~ C~), Q' e
(Hs+ 1H~)
then 7' = Q'-.~P' ~ E. If 7 is a side of Tl 7 cuts R, a-~R in order.If Q~C(S~_~)-(H~K], either 7 already cuts R, a-~R or 7 has endpoints in the dia- metrically opposite sectors (D~H~], [H 1D~) at vt and so can be deformed as required, or if P - H 1 or Q =H~, replace b y P ' ~ (H~ D1), Q'e (D~H~).
Finally if Q~(H~K] the [-expansion of Q begins with a sequence of consecutive R cycles of lengths D .... , D, H beginning with g$_~. Hence P does not begin with an L chain DD ... DH, i.e. P ~ [C~ D~). B u t then either 7 cuts R, a-XR; or 7 has endpoints in the dia-
1 1 8 c. s ~ R ~ s
metrically opposite segments (HzHz_l) , (D 1D2) and we a p p l y L e m m a 3.8; or 7 is not a side of ~ / a n d there are curves 7' close to 7 with endpoints in
(HlHt_l),
(D1D~); or 7 =H?.ID1 and 7 cuts R, a - t R .Now let 7 =Q-.~PEE and suppose we can find a deformation 7* with 7*N R # O . B y Corollary 3.7 either y N R # O or there is a unique region R t with 7 fi R t # O and
R=R~.
I f 7 fi R # O set R @ ) ~ R ; otherwise set R(y ) = R t. I t is clear from L e m m a 3.4 t h a t R(y ) is independent of the deformation 7*"
Suppose 7 =
Q-1. P E E
with no deformation Y* with Y* N R # O , and t h a t 7 is not a geo- desic in ~/. B y Proposition 3.9 we see there are geodesics 7 ' =Q'-~.P'
EE arbitrarily close to 7, with y'* fi R # ~ . We observed above t h a t for a n y region S, S* is a locally constant func.tion of S. Therefore we m a y define R@) = R(?') for 7' close to 7.
I f y E E is a side of ~ , set R@) = R. B y Proposition 3.9, Y cuts
R, a-tit in
succession.I n this case a7 is also a side of ~ / a n d so R(aT) = R. Thus y cuts R@), a-tR(aT) in succession.
Suppose Y E E is not a side of ~ / a n d let Y* be a deformation which cuts
R, (r-lR in
succession. B y L e m m a 3.6 there are regions R1, R 2 so t h a t 7 cuts Rt, R~ in succession ando*(r) R@) = R t b y definition.
R = R 1 or R~ (~), ~ - I R = R 2 or ~2 9
Now ay* cuts R. I f a7 N R # O , R ( a T ) = R . Then 7 cuts R(7), a-~R(aT) in succession.
Otherwise a 7 n
~ = o
b u taT*N R=#O
and a~ na~#O.
ThusR%=aR~
and so R = a(R~(7)). Since o is an automorphism, a(R~(7))=-((rR2) *(~ and thus a7 n a R 2 # ~ and(aR2) *(~r) =R,
which impliesR(aT)=aR 2.
Thus 7 cuts R(7), a-tR(aT) in succession.Finally suppose 7 E Z is not a side of 7/ and is close to a curve ~' which cuts
R@'),
a - t R ( a ~ ') in order. Taking 7' sufficiently close to 7 we haveR(7 )
= R(ff') and R ( ~ ' ) = R(aT').Moreover we m a y assume 7' cuts R(ff'), ~-tR(~y') and so 7 cuts R(7), a-~R(~7).
Now applying Proposition 3.9 to a-17, we m a y find a deformation of a-~7 which cuts R, a - l R in succession, and hence a deformation of 7 which cuts (rR, R in succession. Ap- plying similar reasoning to the above, we see 7 cuts aR(a-tT), R@) in succession. A simple inductive a r g u m e n t and repeated application of L e m m a 3.5 completes the proof of Theo- rem 3.1.
I t is obvious that, for a n y 7 EZ, there is a unique g EE with
gR@)= R.
We shall need a converse to this:P ~ o ~ O S I T X O ~ 3.10.
Let 7 be any geodesic with Y n R=#g]. Then there exist~ a unique g E F so that g7 E ~ and R(gy) = gR.
Proo/.
Suppose g E F is such t h a t g y E Z andR(g?)=gR.
I f R ( g T ) = R , theng=id.
Otherwise,
R(g~)*(~v)=R=g-~R(gT).
Since g is an automorphism,g-l(R(gT)*(ar))=
[g-iR(g7)] *(~),
i.e. R *(~)=g-~R.
Therefore g, if it exists, is unique.S Y M B O L I C D Y N A M I C S F O R G E O D ] ~ S I C F L O W S 119 I f ~ E Z then
R(~)=R
and we m a y take g = i d .So suppose
?=Q-1.P~E.
W i t h o u t loss of generality, we m a y assumePE[C~U1).
I fQ-1.p(~
we m u s t h a v e Qe(HiLk] (see the proof of Proposition 3.9). ClearlyQ~(C~L1],
for then y N R = ~ .Suppose t h a t
Q E(L~L~+I], 1 <~r<k-
1. Arguing as in Proposition 3.9, we see P begins with an R cycle of length at least r, so Pe[Ck_T+~C~). Since r n R~=O, we m u s t have P e[Ck-r+l Ck-~),
the sector at v~+ 1 diametrically opposite (L~L~+I]. Suppose Q ~=L~+I, P ~=C~_~+~.Then b y L e m m a 3.8 we see we can deform ? to obtain a conjugate
R*(~)~=R.
Pick g so t h a tgR*= R.
Now relabel the vertices so t h a tgP~ [C~C~).
Then g? passes to the fight ofgv~+~
and
gP, gQ
arc in diametrically opposite sectors atgv.
Moreovergv~+~
is a vertex of R, and since ~ f~ R* = ~ , g? N/~ = ~ . This forces (with the new labelling),gv~+~ =v~, gPe(D~C1)
andgQe(HzH~).
Now as in the proof of Proposition 3.9,(gQ)-~.gPeZ.
Clearly g? ~ i~ = ~ , so as in Proposition 3.9 there is a unique region R~ withR~(zr)=R
and gy ~ R ~ = ~ , and R ~ =R(gy).
NowR~(Or)=g((g-~R~)*(e)),
since g is an automorphism and thusg-~R=(g-~R~) *~.
B u t
g-lR = R *cry,
therefore b y L c m m a 3.4,g-XR~ = R.
Since R 1 =R(g?), g
is as required.I f either
Q=L~+~ or P=Ce_,+~
we a p p l y the same g as for n e a r b y ? ' and use obvious continuity arguments.Now
if Q =Lx, P e [C~C~)
and ? fi R = ~ , we m u s t have P = C~. Then we m a y take g =id.Finally suppose
Qe(H~+~Hj, 1 <~s<~l-2,
orQe(KHt_~]
ands = l - 1 .
Since ? ~ R ~ we seeP ~ [Dt_~C~).
J u s t as in the proof of Proposition 3.9, this showsQ-1.p ~ .
Thus we m a y t a k e g =id.w 4. Symbolic representation of the geodesic flow
I n this section we show t h a t the geodesic flow on
T~(D/I')
can be represented as a quotient of a special flow over ~, ~; where the height function is the time t a k e n to cross the region R(7 ). We keep the notation and conventions of w 1-w 3.I f ? is an admissible geodesic, let h(?) be the hyperbolic length of ? N R(?). h is infinite if an endpoint of 7 is a cusp. h lifts to a function also denoted b y h on Z. L e t A = { ( e , t): e e l ,
O<~t<h(e)}
a n d let ~0~ be the special flow on A defined b y ~0~(e,t)=
(~ne, t§
when 7 > 0 andO~t+7-Snh(e)<h(~ne)
with a similar definition for 7 < 0 , whereSnh(e)= ~,~-1 haT(e).
cO T
(Notice t h a t ~0 h(a ?) diverges because an arc of ? of finite length can cut only finitely m a n y copies of R.)
L e t yJ~ be the geodesic flow on the unit t a n g e n t bundle M of
D/F,
let M be the unit tangent bundle of D and let ~0: ~ - + M be projection. ~t is geodesic flow on ~ .1 2 0 ~. SERIES
For an admissible geodesic ~, let b ( ~ ) E ~ be the unit t a n g e n t vector pointing along based at the point where ~ enters R(~).
Define II: A ~ M b y
II((e, t)) --
~t(l~b(e)),
where g(e) is the geodesic associated to e. I n what follows we shall frequently identify e and ~(e).
P R O r O S I ~ I O ~ 4.1. I I
is sur]ective,
I I ~ t = ~ t I Iand
:H=II-~(II(e,t))=@z~-~(z~(e)) /or e E E (i.e. H is
1-1except on a set o/the/irst category).
Proo/.
Take u E M. Lift u to 4 E M with the p r o p e r t y t h a t ~ has its endpoint U in R.I f ~ is the geodesic through U in the direction 4, ~, N R ~ O .
B y Proposition 3.10, there is a unique g E F with
gyEE
andR(g~)=gR. g~
is also a lifting of u, and g7 N R ( g T ) ~ . L e t 3 be the hyperbolic distance along g7 from the point V where g~ enters R(gT) togU.
Since U E/~,gU E gR = R(g?).
Then 0 ~<3 <h(g~) (orh(g~)
=0), and gfi =v~b(g~). Also II(gT, 3)=y~(lob(g~)) =p(f~b(gy) =p(g~) =u.
Therefore II is surjeetive.Suppose also II(e, t ) = u , eEE. L e t z~(e)=ft. Then
u=yJtp(b(fl))=p~t(b(fl)).
Thus there is an h E F so t h a thg~=~tb(fl),
and soh-lb(gT)=b(fl).
Thusb(fl)
is the unit tangent vector alongh-lgy
based at the point whereh-lg~
entersh-lR(g~).
This meansh-lgy=fl
andh-lR(g~) = R(fl),
i.e.h - l g R = R(fl).
According to Proposition 3.10,h-lg
is unique and h = id, fl = g~ certainly works. Therefore I I (e, t) = u iff 7~(e) =gy.
Observe z~ is one, two or four-to-one depending on whether g~ has neither, one or b o t h its endpoints in I,J~=0 a - r W .Suppose (e, t)E A, e . . . .
/21/~lele2
.... ~ > 0 andS~h(e)<t § <Sn+lh(e).
Then
b y Theorem 3.1 (3).
Thus
@h(~)b(e)
=a-lb(o'e)
(4.1.1)(ps~h(e) ((~nb(e))
= (ps~h<e) (a~(Pn(e) b (e))
= ~)S~h(e) ( O'n-1 b (ae)) b y (4.1.1)
. . . b ( ( r n e )
(4.1.2)
and
S Y M B O L I C D Y N , ~ M I C S F O R G E O D E S I C F L O W S 121 I I @ J e , t)) = ~h+~_s.~(,)(pb((r~e))
= ~ot+~pO_snn(e)(b((rne))
= y)t+~p(a~b(e))
= y)t+~p(b(e))
= ~ , J e , t).
by|(4.1.2)
A similar computation works for ~ < 0.
We now want to investigate the continuity of H and h. P u t on ~ the usual product topology and metric
n = s u p {m: e, lil < m } .
P R O P O S I T I O N 4.2. ~: ~ + - - S ~ is continuous.
Pro@ I n the no cusp case this follows easily from Property (Ei) of / in w 1, see also the last line of the proof below.
Suppose C is a cusp of R. Suppose the L cycle of generators at C is h I ... hz. Let H = h z ... h 1. Then H ( C ) = C and H ' ( C ) = 1 . B y L e m m a 2.8 of [6],~H acting on S 1 with fixed point C is conjugate b y a M6bius transformation to
acting on R with fixed point 0, with y > 0 . Let
J(H m)
={PES*: P = H -m ...}. One sees easily J ( H m) corresponds to (~(my + 1)-1, 0] for some ~ < 0. Therefore P, Q E J ( H m) ~ ] P - Q l = O(m -I) on S I.Now pick P E S 1 and suppose P corresponds to e = H T ' B 1 H ~ B 2 ... EE + where H~ is a cycle corresponding to a parabolic vertex and B~ is a block containing no such cycles.
Suppose given e > 0.
ml m,~ . . .
Say 3m r so that l/mr <e. Let the length of the sequence H1 BIH2 Br_ 1 be N. Then d(e', e) <2-N~o~Q, a n P e J ( H m,) where Q =~(e'). Also a~, = a~ for 1 <~r<~N and ]a' I ~> 1 on S 1. Therefore [ P - Q I <Ke, for some K depending only on P.
Otherwise, 3 L such t h a t mr<~L, Yr. Thus P ( ~ J ( H r) for a n y parabolic vertex, so akP is a bounded distance away from all the parabolic vertices for each k. Since a'(x) = 1 only at parabolic vertices, this means 3 2 > 1 such t h a t (a~)' ~>jt for all k. Choose N so t h a t 2 -N < e.
If d(e', e) < 2 -N then ~ a~,=ae, k<~N k and so I P - Q I <,~-N.
122 c. SERIES COROLLARY 4.3. ~z: Z-+ S1x S 1
is continuous.
Let ]~*={eEZ: neither endpoint of e on S 1 is a cusp}.
PROPOSITION 4.4.
h is continuous on ~*. I n the no cusp case, h is H5Ider on Y~.
Proo].
We take the no cusp case first.Let 2 be an admissible geodesic in D with endpoints P = e ~~ Q = e 'r Suppose C1, C~ are disjoint geodesics which are cut within bounded arcs by 7" The hyperbolic distance between C 1 and C~ along 7 is a smooth function of 0, ~. Hence if 7' is a geodesic with endpoints
P ' = e ~~ Q'=e ~',
then[ d - d ' I <~K(IO-O' [ + Iq~-q/])
where g depends only on C1, C 2.Let 2 > 1 be the expansive constant for a. Suppose d(7, Y')<2-n- Then ] 0 - 0 ' I ~ 2 -n, R(7 ) always has a vertex in common with R and so is one of a finite number of regions.
Thus h(7 ) is the distance along 7 between a finite number of possible pairs of sides of T/.
Provided 7 does not pass through a vertex of
R(7), I h(7) -h(7')l <--.K7 -~
for K independent of 7.Suppose 7 enters R(7 ) across a geodesic C 1 and leaves across the intersection of C~
and C3. h(7') for 7' near 7 is the distance along 7' from C 1 to one of
C~, C 3.
Both these func- tion are HSlder and their values coincide at 7" Likewise, if 7 coincides with a side of ~/,R(7' )
is one of a finite number of regions meetingR(7 )
and we seeh(7' )
is one of a finite number of HSlder functions all of whose values agree at 7.Now suppose R has cusps. Let K~ be the part of D outside small discs of (Euclidean) radius r round each of the cusps of R.
The above argument shows t h a t h is continuous on geodesics 7 which lie completely inside K~. (Use continuity of the map Z-~ $1• S 1 to replace the constant expansiveness of a.) Now let r-+0.
Now there is a natural topology on A as the suspension of ~ by h.
PROPOSITION 4.3. II: A-->M
is continuous.
Proo/.
I t is enough to see t h a tpb(e)
varies continuously with e E E, and t h a t~ptpb(7)~
pb((17)
ast--> h(7)-.
Now b(7 ) is the unit tangent vector to 7 based at the first intersection S of 7 with the continuous curve aR(7 ). Moreover R(7 ) is locally constant as a function of 7 except when 7 is a side of ~. I n this last case, the appropriate side of
R(7'),
for 7' close to 7, is one of a finite number of continuous curves all of which pass through S.B y Corollary 4.3, the endpoints P, Q of 7 v a r y continuously with e E ~ and clearly 7 varies continuously with P, Q. Hence b(7 ) is a continuous function of e EN.
S Y M B O L I C D Y N A M I C S ~ O R G E O D E S I C F L O W S 123 I f we lift t h e p a t h
~tpb(~)
t olptPb(~)E.~
starting at b(~) w h e n t = 0 , t h e n ast--+h(~)-
t h e base p o i n t of
y~tpb(y)
approaches t h e point T where ? crosses f r o mR(y)
toR(ay).
Therefore limt-~h(r)-yJtb(~)=a-lb(a~). H e n c e
y~tpb(~)-->p(a-lb(a~))=pb(a~)
as required.Remark 4.4.
W e h a v e n o t said a n y t h i n g a b o u t measures on A a n d M. I n [6] we showed there is a n e r g o d i c / r - i n v a r i a n t measure fi on S 1, equivalent to Lebesgue measure, finite in t h e no cusp case a n d infinite otherwise, fi defines a unique a - i n v a r i a n t measure/~ on which projects t o #, b y~(zo ... a,) =~(q(~-~(Za_,...o,))), where
Za
... = { e E Z :er=ar,
Jr] ~<n) a n d O: Z - > Z + is projection.Define a measure v on A b y
[, fh(e)
Jo z .(oeteF,(e)
where Ee = {(e, t) e E : 0 ~< t <
h(e)).
P R O P O S I T I O N 4.5. I I . v
is the natural/low invariant measure on M.
Proo/.
One verifies easily t h a t t h e measure]e ~~ e~r dqJ
on S 1 x S 1 - diagonal is in- v a r i a n t u n d e r t h e n a t u r a l F action. Since a n y geodesic in D is u n i q u e l y determined b y its endpoints on S 1, we can identify T 1 D, t h e u n i t t a n g e n t bundle to D, with (S 1 • S 1 - d i a g . ) • It. T h e measure 2 =[e~~
is i n v a r i a n t u n d e r F acting on t h e left a n d t h e geodesic flow on t h e right.N o w b y Proposition 3.10, a n y u E M has a u n i q u e lifting ~ in T 1D so t h a t t h e geodesic defined b y ~ is admissible a n d ~ has its e n d p o i n t in R(7) (see Proposition 4.1). L e t A__ T 1 D be t h e set of these liftings. I t is clear t h a t 2 [ a (with suitable normalisation) is t h e n a t u r a l flow i n v a r i a n t measure on M. Moreover if q: A - + S 1 • S 1 - d i a g . ,
q-~(7)
has length h(7 ).H identifies
q(A)~_ SI• S 1-diag.
with E. Therefore to see I I . v =2]A, it is e n o u g h t o see t h a t w =]d~
a n d / ~ on E are t h e same. (We can safely ignore t h e sets on which H, ~ are n o t bijective since t h e y are null for all relevant measures.)w is F i n v a r i a n t a n d hence a i n v a r i a n t on
q(A).
I t is clear t h a t w projects to a measure@ equivalent to Lebesgue on E + ( = $ 1 ) , m o r e o v e r @ m u s t be shift i n v a r i a n t on E +.
Therefore ~ a n d fi are shift i n v a r i a n t equivalent measures on E +, a n d fi is ergodie for t h e shift. I t follows t h a t ~
=fi
(if we normalise properly), a n d since ~ determines w u n i q u e l y (just as fi determines #), we are done.Notice t h a t fi is t h e Gibbs measure corresponding to t h e function
-log l/'(x)[
on S L124 c.s~RI~S
I t n o w follows from t h e symbolic representation t h a t t h e geodesic flow is ergodie (since t h e shift a on E is). I n t h e c o m p a c t case we can deduce t h e flow is Bernoulli. One needs t o k n o w t h e flow is K; this is a general fact, see for e x a m p l e [17]. T h e result follows f r o m T h e o r e m 4.3 of [16], (a K-flow which is t h e special flow over a shift u n d e r a l~61der continuous function is Bernoulli). (One makes an obvious modification to deal with t h e f a c t the height function m a y vanish, since 3 N such t h a t h(e)+ ... +h(o~e)>~ c > 0 , Ve E E.) W e hope to investigate t h e n o n - c o m p a c t case elsewhere. (The flow is k n o w n t o be Bernoulli in this case also, see [7].)
w 5. Quasi-conformal deformations
T h r o u g h o u t w 1-w 4, we a s s u m e d t h a t F h a d a f u n d a m e n t a l region R which satisfied t h e p r o p e r t y (*). I n [6] we showed t h a t if F ' is a n y F u c h s i a n g r o u p of t h e first kind, t h e n there is a group F satisfying (*), such t h a t there is a quasi-conformal deformation ?': F - ~ F ' . W e n o w show h o w to use this deformation to c a r r y over t h e results a b o v e to t h e general c a s e .
W e first summarize t h e facts we need a b o u t quasi-conformal maps. F o r details, see [4].
(I) There is an isomorphism j: F ~ F ' , a n d a diffeomorphism ~o~: D ~ D ' = D so t h a t
](g)
= (9/tg((9/~) -1, g E F .(2) (o~ restricts to a h o m e o m o r p h i s m h: SI-+S 1 so t h a t h(gx)=j(g)h(x), x E S 1, g E P . h is t h e so-called boundary m a p of w~.
(3) I f ~ is a geodesic in D, t h e n 7' = eel(7) is a so-called quasi-geodesic in D ' . There is a unique geodesic ~ in D ' with t h e same endpoints as o)~(~), ~ is a b o u n d e d hyperbolic distance from co~(?) (with b o u n d depending only on co~), [13].
Notice t h a t if ~, fl are geodesics in D t h e n ~ n f l ~ O if a n d only if ~ n fi=~O.
L e t a be a geodesic in D which is an edge of T/, a n d let v be a v e r t e x of 7 / o n a. L e t fil ... fir be t h e other edges of ~ t h r o u g h v. T h e n ~ n f i ~ O , 1 <~ i <~r, b u t these intersec- tions m a y all be distinct points. L e t a(v) = {~ fi fi,}[=l. L e t w be a v e r t e x of 7 / a d j a c e n t to v along a. T h e n if 7 is a n y o t h e r edge of ~ t h r o u g h w, ~ N f i , = O , i <~i~r, a n d so we can find disjoint closed intervals I~(v), I~(w) on a so t h a t a(v)_~Int I~(v), a(w)_~Int I~(w).
More generally if {v,}~_~ are t h e vertices of T / a l o n g a in order t h e n there are disjoint closed intervals {I~(v,)}~ _~o along ~ in t h e same order as {v,}, a(v,)_~ I n t Ia(v,).
L e t Q(v) be t h e open convex hull in D' of t h e set {Ia(v): ~ is an edge of T / t h r o u g h v}.
N o w let t 1 ... t n be t h e sides of a c o p y S of R in D. Since n o n - a d j a c e n t sides of S do
S Y M B O L I C D Y N A M I C S F O R G E O D E S I C F L O W S 125 not meet, the same is true of [1 ... i~ and thus tl, .-., tn bound a closed polygonal region in D'. Let
Q ( S ) = ~ -
U {Q(v): v is a vertex of S} and letQ(D)=
D ' - U {Q(v): v is a vertex of~}.
If we collapse each of the regions
Q(v)
to a point we obtain a net Q(~) whose sides are the portions of the edges 0~ outside the regionsQ(v)
and which is topologically identical with the net ~.Now let ~) be a geodesic in D'. We say ~) passes across
Q(v) if 2 N Q(v) ~ .
Let the sides of T/meeting at v be t 1 ... t2k , going in clockwise order round v. Moving clockwise roundQ(v)
one cuts successively ~1 ... i~. Let ~) cut~Q(v) in
points P, Q. Letfl(v)
be the arc of~Q(v)
joining P to Q which cuts the smaller number of sides [~. (If both arcs cut k or k + 1 sides choose fl to be the arc passing to the left ofQ(v).)
Now let ~ be the curve obtained from p b y replacing p with [~) -Q(v)] U
fl(v)
in a neigh- bourhood ofQ(v),
for every vertex v. I n the collapsed net Q(~), ~ becomes a curve Q(7) which passes through a vertex v whenever p ~Q(v) ~-~.
THV.ORV,~ 5.1.
Let ~ be a geodesic in D" corresponding to an admissible geodesic y in D.
We can/ind a distinguished region Q(S(y)) such that
(1)~ h Q(S(y)):~O
(2) ~ n
Q(S(~)):~o ~s(~) =R
(3) ~
cuts in succession Q(S(y)), (~-IQ(S((~)) ...
Proo/.
The idea is obviously to imitate w 3. We define what is meant by a curve inQ(D)
passing near a vertex of Q(~/) just as in w 3. L e m m a 3.4 depends only on the topology of ~ and the position of the endpoints of y relative to ~; and thus carries over to Q(~) and ~. To prove Lemma 3.5, it is enough to see t h a t ~ is geodesically convex, or equivalently t h a t the interior angles of ~ are less t h a n g. B u t a vertex of ~ is formed by the intersection of two geodesics with distinct endpoints, and therefore the angle between a n y adjacent pair of sides is less than z.The proofs of Lemma 3.6 and Corollary 3.7 are unchanged. Lemma 3.8 and Proposi- tion 3.9 again depend only on topological properties of ~ and the position of the endpoints of y. The rest of the proof is as in w 3.
We shall say a permutation ~ of Z 'acts on finite blocks' if there are integers ... < n 1 < n 2 < ... such t h a t g maps each interval n~ ~<r <nt+ 1 onto itself. The importance of this will be t h a t we can keep track of a 'base point' on a sequence, b y choosing the left endpoint of some fixed block to be the base point. If we require permutations to preserve a base point, the sequence n _la), n_l(2 ) .... uniquely determines ~.
