Table Of Content5 A divergent Teichmu¨ller geodesic with
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uniquely ergodic vertical foliation
2
n
a
J Yitwah Cheung
9
Northwestern University
1
Evanston, Illinois
]
S
email: [email protected]
D
.
h
and
t
a
m
Howard Masur
[ ∗
University of Illinois at Chicago
1
v
Chicago, Illinois
6
9 email: [email protected]
2
1
0 February 1, 2008
5
0
/
h
t
a
Abstract
m
: We construct an example of a quadratic differential whose vertical
v
i foliation is uniquely ergodic and such that the Teichmu¨ller geodesic
X
determined by the quadratic differential diverges in the moduli space
r
a of Riemann surfaces.
1 Introduction and Statement of theorem
We let S be a closed surface of genus g 2 and T the Teichmu¨ller space of S
g
≥
equipped with the Teichmu¨ller metric. Let Map = Diff+(S)/Diff (S) denote
0
the mapping class group of S. It acts on T by isometries with quotient space
g
∗This researchis partially supported by NSF grant DMS0244472.
1
, the moduli space of surfaces of genus g. It is well-known that is not
g g
M M
compact; one leaves compact sets of by finding curves whose lengths in
g
M
the hyperbolic metric on the surfaces goes to zero. For any Riemann surface
X T a holomorphic quadratic differential q = q(z)dz2 assigns to each
g
∈
uniformizing parameter z a holomorphic function q(z) such that q(z)dz2 is
invariant under holomorphic change of coordinates. Away from the zeroes of
q therearenaturalcoordinatessothatq(z) 1sothatq definesametric dz2
≡ | |
which is locally Euclidean away from the zeroes of q. A saddle connection γ
is a geodesic in this metric joining a pair of (not necessarily distinct) zeroes
that has no zeroes in its interior. It is represented by a straight line in the
natural Euclidean metric and determines a holonomy vector with horizontal
and vertical components which we denote by λ(γ) and v(γ) respectively.
Associated to q are the horizontal and vertical trajectories. These are
the arcs along which q(z)dz2 > 0 and q(z)dz2 < 0. For each t R one
∈
can define a new quadratic differential g (q) on a new Riemann surface X
t t
by expanding along the horizontal trajectories of q by a factor of et and
contracting along the vertical trajectories by a factor of e−t. The family of
X form a Teichmu¨ller geodesic through X whose projection to defines a
t g
M
geodesic in M . The underlying map X X of Riemann surfaces is called
g t
→
a Teichmu¨ller map. In the appropriate natural coordinates z = x + √ 1y
−
on X away from the zeroes of q, and natural coordinates z = x +√ 1y on
t t t
−
X away from the zeroes of g (q), the map is given by
t t
x = etx,y = e−ty.
t t
One may study the dynamics of the family of vertical trajectories of q
which defines the vertical foliation F . An important notion in topological
q
dynamics is that of minimality. The foliation F is minimal if the full orbit
q
of every leaf is dense. It is a standard fact [St] that if there are no saddle
connections with zero horizontal holonomy, then the vertical foliation F is
q
minimal.
For minimal foliations it is natural to study their ergodic behavior. A
foliation is uniquely ergodic if every leaf is uniformly distributed on the sur-
face. Equivalently, the foliation is uniquely ergodic if there is a unique, up
to scalar multiplication, measure transverse to F invariant under holonomy
q
along leaves of F . (To avoid trivialities, we assume these measures are sup-
q
ported on the complement of the singularities) An interesting phenomenon
found by [S], [V1], [K],[KN] is the existence of minimal foliations that are
2
not uniquely ergodic. The last two examples were in the context of interval
exchange transformations, but a suspension of an interval exchange transfor-
mation yields a (oriented) measured foliation.
In [M] a connection was made between the dynamics of the foliation F
q
and the dynamics of the corresponding geodesic X determined by q in .
t g
M
It was shown that if the foliation is minimal, but not uniquely ergodic, then
the geodesic diverges in . This means that it eventually leaves every
g
M
compact set. The question arises if this condition is necessary; does X
t
divergent imply F is nonuniquely ergodic? We show the answer is negative.
q
Theorem 1. There exist quadratic differentials q with uniquely ergodic F
q
such that the Teichmuller geodesic X diverges in .
t g
M
We will construct an example in genus two, although our method will
produce examples in any genus. That such examples should exist was al-
ready suggested by Kerckhoff to the second author (oral communication).
We consider the “zippered rectangle” construction of Veech [V2]. This is a
way of defining an Abelian differential on a Riemann surface such that the
first return map of the vertical trajectories of the Abelian differential to a
horizontal interval is a given interval exchange transformation.
We then recall the notion of Rauzy induction ([R].) This is a procedure
whichbytakingthefirstreturnmaponasubintervalofthehorizontalinterval
gives a new interval exchange and also gives a new zippered rectangle. Asso-
ciated to Rauzy induction is a graph of permutations of interval exchanges.
The pointof thisconstruction inour context isthat successive applications of
Rauzy induction gives a discrete set of points along the Teichmu¨ller geodesic
defined by the zippered rectangles.
We then specialize to interval exchanges on four intervals and explicitly
find the graph of permutations. We produce an infinite path in this graph
that allows us to explicitly compute the lengths and heights of the zippered
rectangles given by Rauzy induction. The interval exchange whose sequence
of Rauzy inductions give this path has the desired properties. Namely, it
is uniquely ergodic, and any Abelian differential formed by the zippered
rectangle construction yields a divergent geodesic.
3
2 Interval exchanges and zippered rectangles
Let λ Rm be a vector of positive lengths and π an irreducible permutation
∈ +
on 1,...,m for some m 2. Recall irreducibility means π 1,...,k =
{ } ≥ { } 6
1,...,k for1 k < m. Associatedtothepair(λ,π)isanintervalexchange
{ } ≤
map T defined as follows. Let I = [β ,β ) where β = 0 and β =
j j−1 j 0 j
λ + +λ for j = 1,...,m. Then T is the map defined on I = I by the
1 j j
··· ∪
formula: for x I
j
∈
T(x) = x λ + λ .
i πj
−
Xi<j πXi<πj
We recall a construction in [V2] of “zippered rectangles” to suspend T.
Let h,a Rm be vectors satisfying the inequalities
∈
h > 0 (1 j m) (1)
j
≤ ≤
0 < a min(h ,h ) (1 j < m,j = π−1m) (2)
j j j+1
≤ ≤ 6
0 < aπ−1m hπ−1m+1 (3)
≤
hπ−1m am hm (4)
− ≤ ≤
and the system of linear equations
h a = h a (0 j m) (5)
j j σj+1 σj
− − ≤ ≤
where a = h = h = 0 by convention and σ is the permutation on
0 0 m+1
0,1,...,m defined by
{ }
π−11 1 j = 0
−
σj = m j = π−1m . (6)
π−1(πj +1) 1 otherwise
−
The surface M(π,λ,h,a) is obtained by glueing together m rectangles
along their boundaries in the following manner. Let R be the rectangle in C
j
with base I on the real axis and height h . There are three cases depending
j j
on the sign of a and in each case there will be three sets of identifications,
m
all of which are the form z z +c.
→
First, consider the case a = 0. The first set of identifications is
m
(1) the top of R is glued to the interval TI at the base for j = 1,...,m.
j j
4
To describe the remaining identifications we shall use the notation
R+[a,b] R−[c,d]
j ∼ k
to mean the right side of R between heights a and b is glued to the left side
j
of R between heights c and d. (For this to be well-defined, we must have
k
b a = d c.) The remaining identifications in the case a = 0 are
m
− −
(2) R+[0,a ] R− [0,a ] for j = 1,...,m 1, and
j j ∼ j+1 j −
(3) R+[a ,h ] R− [a ,h ] for j = 1,...,m.
j j j ∼ j+1 σj σj+1
Now consider the case a > 0. Let j = π−1m. All identifications remain the
m
same except the jth in (2), which is replaced with
R+ [0,h ] R− [0,h ] and
π−1m j ∼ π−1m+1 j
R+[0,a ] R− [h ,a ].
m m ∼ π−1m+1 j j
Similarly, in the case a < 0 all identifications remain the same as in the
m
first case except the mth in (3), which is replaced with
Rπ+−1m[aπ−1m,hπ−1m] ∼ Rσ−m+1[aσm,hσm+1 −hm] and
R+[0,h ] R− [h h ,h ].
m m ∼ σm+1 σm+1 − m σm+1
The collection of glued rectangles M = M(π,λ,h,a) is called the zippered
rectangle associated to (π,λ,h,a). Since the glueing maps are of the form
z z +c, M carries the structure of a Riemann surface and the 1-form dz
→
induces anAbeliandifferentialω = ω(π,λ,h,a)onM. The intervalexchange
T isthefirstreturnmaptoI undertheflowintheverticaldirectiongenerated
by the vector field ∂/∂y.
Note if each cycle of σ contains at least two elements in 1,...,m 1 ,
{ − }
then ω has a zero at (0,0) and at λ + ...+ λ + ia for j = 1,...,m. In
1 j j
particular, each R has at most one zero on each of its vertical sides. More
j
specifically, the left side always contains a zero while the right side contains
a zero except when j = m, a < 0 or when j = π−1m, a > 0.
m m
Example 1. Let m = 4, πj = 5 j and λ R4. Then (π,λ,h,a) is a
− ∈ +
zippered rectangle with h = (1,3,3,2), a = (2,2,2,1) or with h = (2,3,3,1)
and a = (1,1,1, 1), where σj = j 2 (mod 5). See Figure 1.
− −
5
c a d c d
c a c
b d b b
a
a b d
a = b > 0 a = −b < 0
4 4
Figure 1: Zippered rectangles in Example 1.
3 Rauzy induction
Let denote an alphabet with m 2 elements. A marked interval exchange
A ≥
is a pair (T,ν ) where T is an interval exchange on m intervals and ν a
0 0
bijection from 1,...,m to . We think of the intervals of T as being
{ } A
markedfromlefttoright withnames ν (1),...,ν (m). IfT isa(λ,π)interval
0 0
exchange, then the names of the image intervals from left to right are given
by ν (1),...,ν (m) where
1 1
ν = ν π−1.
1 0
◦
An alternative way to represent a marked interval exchange is via a triple
(λ,ν ,ν ) where λ : R is the function given by
0 1 +
A →
λ(ν (j)) = λ for j = 1,...,m.
0 j
The function λ and the vector λ Rm are represented by the same symbol
∈ +
so that λ(α) is the length of the interval marked α, while λ is the length of
j
the jth interval from left to right. We shall also refer to the pair (ν ,ν ) as
0 1
a marked permutation. In what follows, we fix = 1,...,m .
A { }
We define Rauzy induction on the space of marked interval exchange
transformations. First for λ a vector let λ denote the sum of the lengths of
| |
the components. Now given a marked interval exchange (λ,ν ,ν ), consider
0 1
the first return map found by inducing on the longer of the two subintervals
[0, λ λ(ν (m))) where i 0,1. If
i
| |− ∈
Case (a) λ(ν (m)) > λ(ν (m))
1 0
6
then the interval marked ν (m) is shortened. The new marked permutation
1
is (ν′,ν′) where ν′ = ν and ν′ is the ordering obtained from ν by inserting
0 1 1 1 0 0
ν (m) is the position immediately after ν (m) and moving forward one place
0 1
the names of the intervals appearing after ν (m).
1
Likewise, if
Case (b) λ(ν (m)) > λ(ν (m))
0 1
then the interval marked ν (m) is shortened, ν′ = ν and ν′ is the ordering
0 0 0 1
obtained from ν by inserting ν (m) in the position immediately after ν (m)
1 1 0
and moving forward one place the names of the intervals appearing after
ν (m). We ignore the case λ(ν (m)) = λ(ν (m)).
0 0 1
TheresultofRauzyinductionisanewmarkedintervalexchange(λ′,ν′,ν′).
0 1
In the first case the lengths are related by
λ = Aλ′, (7)
whereAistheelementary matrixwhose onlynonzerooff-diagonalentryisa1
in the (ν (m),ν (m)) position. In the second case the matrix is the transpose
1 0
of the above matrix so that it has a 1 in the (ν (m),ν (m)) position. We
0 1
can rephrase the statement about the new marked permutation as follows.
The new marked permutation is a(ν ,ν ) in the first case and b(ν ,ν ) in
0 1 0 1
the second where a,b are the bijections on the set of marked permutations
defined by
a(ν ,ν ) = (ν c ,ν ) k = ν−1(m) (8)
0 1 0 ◦ k 1 1
b(ν ,ν ) = (ν ,ν c ) l = ν−1(m) (9)
0 1 0 1 ◦ l 0
and
i i k
≤
c (i) = m i = k +1 . (10)
k
i 1 k +1 < i m
− ≤
(Thecontext willmake itclearwhether aisanoperationontheset ofmarked
permutations or a vector of lengths in Rm.)
By repeating this procedure, we obtain a sequence of elementary matrices
(A ) and length vectors (λ ) satisfying
k k≥1 k k≥1
λ = A A λ .
1 k k
···
7
The sequence A A is called the expansion of λ. The corresponding
1 k
··· ···
sequence of marked permutations is a path in the extended Rauzy diagram.
ThisisadirectedgraphwhosevertexsetistheextendedRauzyclassR(ν ,ν )
0 1
consisting of all marked permutations obtainable by applying the operations
a and b to (ν ,ν ). There is a directed edge from x to y if and only if either
0 1
y = ax or y = bx. The edges in the Rauzy diagram are labeled by a or b as
the case may be. A directed path starting at (ν ,ν ) is uniquely represented
0 1
by a word in thealphabet a,b . If w is a word in a,b , let x;w denote the
{ } { } h i
path starting at x obtained by following the letters of w from left to right.
Let [x;w] denote the corresponding product of elementary matrices. If the
path ends at y and w′ is another word, then [x;ww′] = [x;w][y;w′] where
ww′ denotes the concatenation of the words w and w′.
3.1 Rauzy induction with heights
Rauzy induction also gives a transformation on the space of zippered rect-
angles. Let M(π,λ,h,a) be a zippered rectangle. Let
h′ = Ath (11)
and define, if λm < λπ−1m (edge a)
a j < π−1m
j
a′j = hπ−1m +am−1 j = π−1m (12)
a π−1m < j m
j−1
≤
and if λm > λπ−1m (edge b)
a j < m
a′ = j . (13)
j (cid:26) (hπ−1m aπ−1m−1) j = m
− −
Then the Riemann surface M(π,λ,h,a) is biholomorphically equivalent to
M(π′,λ′,h′,a′) and the Abelian differential ω(π,λ,h,a) is equivalent to the
Abelian differential ω(π′,λ′,h′,a′). In the first case, the last rectangle is
stacked on top (as far right as possible) of a piece of the one that goes to the
last. In the second case a piece of the last rectangle is stacked on top of the
one that goes to the last. See Figure 2. In particular, the last rectangle of the
new set of zippered rectangles has a zero on its right side if the corresponding
edge is labeled with an ’a’ (the first case).
8
c a d c c a d c
4 1
b d b d
e e e e
a b a b
λ(1) > λ(4) λ(4) > λ(1)
(a) (b)
Figure 2: Rauzy induction with (π,λ,h,a) as in Example 1. The solid line
indicates where a piece of the last rectangle is stacked on top of the first.
Note that the name of the rectangle with a new height is 4 in the first case
and 1 in the second.
4 Interval exchanges with four intervals
For the rest of the paper we will consider interval exchanges on four intervals.
Let π = (ν ,ν ) where ν (j) = j and ν (j) = 5 j. The extended Rauzy
0 0 1 0 1
−
diagram R(π ) is shown in Figure 3.
0
Now for each n we form a pair of paths
π ;babnab2
0
(cid:10) (cid:11)
and
π ,abanba2
0
(cid:10) (cid:11)
that begin and end at π . We form the corresponding product of elementary
0
matrices A = [π ;babnab2] and B = [π ;abanba2] and set C = A B . One
n 0 n 0 n n n
computes C as follows. At each stage on the path, the marked permutation
n
(ν′,ν′) is determined from the previous marked permutation (ν ,ν ) by (8)
0 1 0 1
and (9). Each marked permutation determines an elementary matrix as in
(7) and the discussion that follows. The matrix C is the product of these
n
elementary matrices. It is then easily verified that the entries of C are
n
9
b a
a
b
a b
a b
π
0
b a
b a
a b
Figure 3: Rauzy diagram for the marked permutation π .
0
polynomials in n whose leading terms are
1 1 1 1
1 2n 2n 1
.
n n2 n2 0
2 n n 2
For each n, set
n
H = C
n j
Yj=1
For any λ H R4, let λ be such that
∈ n + n
λ = H λ .
n n
Then successive applications of Rauzy induction on the interval exchange
(λ,π ) produces the interval exchange (λ ,π ). Now let Σ be the standard
0 n 0
simplex in R4 and regard each C as a projective linear transformation from
j
Σ to itself.
Proposition 1. The sets Σ = H Σ are a decreasing sequence with infinite
n n
intersection a single point denoted by λ . The interval exchange (λ ,π ) is
0 0 0
uniquely ergodic.
10
