Slowly divergent geodesics in moduli space

Slowly divergent geodesics in the moduli space of Riemann surfaces of genus at least 2 are constructed via cyclic branched covers of the torus. Nonergodic examples (i.e. geodesics whose defining quadratic differential has nonergodic vertical foliation) diverging to infinity at sublinear rates are constructed using a Diophantine condition. Examples with an arbitrarily slow prescribed growth rate are also exhibited.


Introduction
Let M g denote the moduli space of closed Riemann surfaces of genus g 2, endowed with the Teichmüller metric τ . A geodesic in M g is determined by a pair (X 0 , q) where X 0 is a Riemann surface and q is a holomorphic quadratic differential on X 0 . The differential q defines a flat metric with isolated singularities on X 0 together with a pair of transverse measured foliations defined by q > 0 (the horizontal) and q < 0 (the vertical). By a theorem of Masur [Ma92] the vertical foliation of q is uniquely ergodic if X t accumulates in M g as t → ∞. Therefore, a nonergodic geodesic, by which we mean a geodesic determined by a pair (X 0 , q) such that the vertical foliation of q is not uniquely ergodic, must eventually leave every compact set. A geodesic with this latter property is said to be divergent. The original motivation of this study is to answer a question of C. McMullen regarding the existence of slowly divergent nonergodic geodesics: The examples are realized using branched covers of the torus satisfying a Diophantine condition. Let (X, q) be the g-cyclic branched cover of T = (C/Z[i], dz 2 ) obtained by cutting along an embedded linear arc γ. (See §2 for a precise definition.) Each θ ∈ S 1 determines a Teichmüller geodesic X θ t in M g starting at X θ 0 = X. A direction θ is also said to be nonergodic, divergent, slowly divergent, etc. if the corresponding geodesic X θ t has the same property. For a slowly divergent direction it makes sense to consider the sublinear rate: A pair (x 0 , y 0 ) ∈ R 2 is said to satisfy a Diophantine condition if there are constants c 0 > 0 and d 0 > 0 such that for all pairs of integers (m, n) ∈ Z 2 \ {0} inf l∈Z |mx 0 + ny 0 + l| > c 0 max(|m|, |n|) d0 . ( Let x 0 + iy 0 ∈ C be the affine holonomy γ dz of γ. Theorem 1. If (x 0 , y 0 ) satisfies a Diophantine condition with exponent d 0 then for every e 0 > max(d 0 , 2) there is a Hausdorff dimension 1/2 set of slowly divergent nonergodic directions θ with sublinear rate r + (θ) 1 − 1/e 0 .
It should be emphasized that Theorem 1 does not give any examples of nonergodic directions with r + (θ) 1/2. In fact, after this paper had been accepted, it was shown that if r + (θ) 1/2 then θ is uniquely ergodic. See [CE].
As a complement to Theorem 1, we also prove Theorem 2. If (x 0 , y 0 ) ∈ Q 2 then there are directions which are divergent with an arbitrarily slow prescribed rate, i.e. given any function R(t) with R(t) → ∞ as t → ∞ there exists a divergent direction θ such that τ (X θ t , X θ 0 ) R(t) for all sufficiently large t.
In M 1 , the asymptotic behavior of a geodesic is determined by the arithmetic properties of its endpoint in R∪{∞}. For example, (1) holds iff the endpoint is a Roth number: for any ε > 0, there exists c 0 > 0 such that for any p, q ∈ Z, |α − p/q| > c 0 /|q| 2+ε . The question asked by C. McMullen was inspired by a recent result of Marmi-Moussa-Yoccoz concerning interval exchange maps, which give another a source of Teichmüller geodesics in M g . In [MMY], they construct examples of uniquely ergodic interval exchange maps based on a certain "Roth type" condition, which is apparently stronger than (1). Theorem 1 shows that the condition (1) alone is not sufficient to ensure unique ergodicity. suggestions that significantly improved the exposition of the paper. In addition thanks must go to Howard Masur, Curt McMullen and Barak Weiss for many enlightening discussions on the topic of slow divergence. Last, but not least, the author is indebted to his wife Ying Xu for her constant and unwavering support.

Cyclic branched covers along a slit
The g-cyclic branched cover of T along γ is defined as follows. Endow the complement of γ with the metric defined by shortest path and let T ′ be its metric completion. T ′ is a compact surface with a single boundary component and is known as a slit torus, i.e. T slit along γ. Let γ ± denote the two lifts of γ under the natural projection T ′ → T which maps ∂T ′ onto γ. For convenience, we assume γ is defined on the unit interval. Let X be the quotient space of T ′ × Z/gZ obtained by identifying (γ − (t), n) with (γ + (t), n + 1) for all t ∈ [0, 1] and n ∈ Z/gZ. The map π : X → T induced by projection onto the first factor is a branched cover of degree g, holomorphic with respect to a unique complex structure on X. The pair (X, π * dz 2 ) is called the g-cyclic cover of T = (C/Z[i], dz 2 ) along γ.
Note that X is a closed Riemann surface of genus g. The map π is branched at two points corresponding to the zeros of the quadratic differential π * dz 2 . Each branch point lies over an endpoint of γ.

Teichmüller geodesics and saddle connections
The Teichmüller geodesic X θ t will be described explicitly. Let g θ t : R 2 → R 2 be the linear map which contracts distances by a factor of e t/2 in the θ direction while expanding by e t/2 in the direction perpendicular to θ. There is an atlas of charts {U α , ϕ α } covering X away from the branch points such that dϕ α = π * dz. The complex structure of X is uniquely determined by this atlas. It is easy to check that {U α , g θ t •ϕ α } defines a new atlas of charts uniquely determining a new complex structure on X. (Here, we used the standard identification C = R 2 .) The space X with this new complex structure is the Riemann surface X θ t referred to in the introduction. The family X θ t defines a unit speed geodesic in M g with respect to the Teichmüller metric τ . It carries a quadratic differential q θ t which is the square the holomorphic 1-form determined by the new charts.
A saddle connection is a geodesic segment which joins a pair of branch points without passing through one in its interior. Associated to an oriented saddle connection α in X is a complex number α π * dz which we identify with the corresponding vector in R 2 . The collection of vectors associated to saddle connections in X will be denoted by V .
Let W = ±(x 0 , y 0 ) + Z 2 and Z = {(p, q) ∈ Z 2 : gcd(p, q) = 1} and note that Indeed, a saddle connection in X projects to a path in T whose lift to R 2 lies in W ∪ Z. Conversely, (3) implies {x 0 , y 0 , 1} is independent over Q so that the slope of any vector in W ∪ Z is irrational. Hence, any vector in W ∪ Z can be represented by a geodesic arc in T which joins the endpoints of γ without passing through either one. The lift of this arc to X is a saddle connection. Note that the set of vectors associated with saddle connections in (X θ t , q θ t ) is simply given by g θ t V . For any discrete subset S ⊂ R 2 , let ℓ(S) denote the length of the shortest vector in S. To control distances in M g , we need the following result which is proved in slightly greater generality in [Ma93].
Remark 2.2. The square in (4) does not appear in [Ma93] due to a different normalisation of the Teichmüller metric. In our case, the sectional curvature along Teichmüller disks is −1, instead of −4.

Summable cross products condition
The surface X carries a flat metric induced by π * dz so that it makes sense to talk about parallel lines, area measure, etc. For any θ ∈ S 1 , let F θ denote the foliation of X by lines parallel to θ. The foliation F θ is ergodic (with respect to area measure) if X cannot be written as a disjoint union of two invariant sets of positive measure. (An invariant set is one that can be written as a union of leaves.) By definition, θ is a nonergodic direction iff F θ is not ergodic. The next lemma will be useful for finding nonergodic directions.
Lemma 2.3. Let π ′ : X ′ → T be the g-cyclic branched cover of T along another arc γ ′ with the same endpoints as γ. Then π ′ is biholomorphically equivalent to π if and only if γ − γ ′ represents the trivial element in H 1 (T, Z/gZ).
Proof. Let U ⊂ X be the set of points lying over the complement of γ in T and let U ′ ⊂ X ′ be defined similarly. We shall identify a dense subset of U with a dense subset of U ′ as follows. Fix a base point z 0 ∈ γ ∪ γ ′ and let U be the set of paths in X starting at z 0 which are transverse to π −1 γ. For any α ∈ U, the intersection number i γ (α) ∈ Z/gZ is the number of times α crosses γ positively. (This notion depends on a choice of orientation for T , which we assume has been fixed.) The map α → (α(1), i γ (α)) (where α(1) denotes the terminal point of α) induces a bijection between U and U/ ∼ where Similarly, U ′ may be identified with classes of paths transverse to γ ′ using the above with γ replaced by γ ′ . Note i γ (α−α ′ ) = i γ ′ (α−α ′ ) iff the homology intersection of the cycles γ −γ ′ and α − α ′ vanishes. Therefore, if γ is homologous to γ ′ , there exists a bijection of U ∩ π −1 π ′ (U ′ ) with U ′ ∩ π ′−1 π(U ) which extends uniquely to a biholomorphic equivalence between π and π ′ . Conversely, if γ is not homologous to γ ′ , then there is a closed curve β disjoint from γ ′ such that i γ (β) = 0. Since its lift is closed in X ′ but not in X, π ′ cannot be equivalent to π.
In the sequel, the cross product of two vectors in R 2 is defined to be a scalar a, b × c, d := ad − bc.
Lemma 2.4. Let (w j ) j 0 be a sequence of vectors of vectors in ± x 0 , y 0 + gZ 2 whose Euclidean lengths form an increasing sequence and suppose that ∞ j=0 |w j × w j+1 | < ∞. (5) Then |w j | −1 w j converges to a nonergodic direction in X.
Proof. Note that the direction of the vector x 0 , y 0 associated to γ is nonergodic because π −1 γ partitions X into g invariant sets of equal area. Similarly, there is associated to each w j a g-partition of some branched cover of T that is biholomorphically equivalent to π, by Lemma 2.3. Since a biholomorphism preserves partitions by invariant sets, it follows that the direction of each w j is also nonergodic. Now observe that the symmetric difference of the g-partitions of X associated to a consecutive pair of vectors in the sequence is a union parallelograms whose area is bounded above by a constant times the absolute cross product. An elementary argument 1 shows lim |w j | −1 w j exists while (5) implies the sequence of g-partitions converge measure-theoretically to a g-partition invariant in the limit direction.
Remark 2.5. Lemma 2.4 is due to Masur-Smillie in genus 2 and is the precursor to a general criterion for nonergodicity developed in [MS]. Their original motivation was to give a geometric interpretation, in the context of rational billiards, of certain Z/2 skew-products studied by Veech in [Ve].

Analysis of the shortest vector function
The main result of this section is Proposition 3.6, which is used to control rates. Its hypotheses are motivated by Lemmas 3.3, 3.4 and 3.5 while its conclusion is motivated by Lemmas 3.1 and 3.2.
1 For more details see the proof of Lemma 1.1 in [Ch].
Proof. From (8) we see the unique solution to |g θ t v| = |g θ t v ′ | is determined by The second and third hypotheses imply t is well-defined and that e 2t > 1 iff |v| < |v ′ |. Hence, the first hypothesis implies t > 0. Note that the second and third hypotheses hold after v and v ′ are replaced by the vectors g θ t v and g θ t v ′ . Since these vectors have the same length, an elementary calculation shows the sine of the angle φ between them is at least 1/3. (Note this is the sine of the angle between √ 2, 1 and 1, √ 2 .) Since g θ t preserves cross products, we have |v × v ′ | = e −m sin φ, which implies (15). To get (14) we first consider the unique maximum time T ′ of the function t → − log |g θ t v ′ | 2 . The analog of (11) for T ′ is Note that t T ′ for the second hypothesis together with (16) and (17) implies Now using the definition of m, the analog of (8) for v ′ , (16) and (17) we have (14) now follows from (15).
In fact, the condition (18) Suppose u ∈ V is a vector with |g θ T u| |g θ T v|. Since g θ T stretches Euclidean lengths by a factor of at most e |T |/2 , we have |u| εe |T |/2 √ 2|v| by the above equations. Observing that g θ T preserves cross products, we have This proves the first part of the lemma while the second part follows by discreteness of V .
Lemma 3.4. For any w ∈W there exists a unique v ∈ Z up to sign such that Proof. Since W ∩ Z = ∅, the hypothesis implies the minimum exists. In fact, it must be realized by some vector in Z, for if w ′ ∈ W then w ′ = w + du for some u ∈ Z and positive integer d so that |w × w ′ | = |w × du| |w × u|. Now let v be the vector associated to w and consider the capped rectangle R defined by the inequalities |u| √ 2|w| and |w × u| |w × v|. It is enough to show R ∩ Z = {±v}. Suppose there exists u ∈ R ∩ Z such that u = ±v. Note that the area of R is < 4 √ 2|w × v| 2 while the area of the parallelogram P with vertices at ±u and ±v is exactly 2. This is absurd since P ⊂ R. Therefore, For any w ∈W define where v is the vector given by the Lemma 3.4.
then there exists a piecewise linear function Λ(t) satisfying and whose critical points are given by where j > j 1 for some j 1 > 0.
Since v ′ is the vector associated to w j+1 , we note here that for any θ ∈ I(w j+1 ) and j large enough Next, by Lemma 3.5, whose hypothesis ε 1/5 is implied by (iii) and |w j+1 | (g + 1)|v ′ | (from (i)), we have I(w j+1 ) ⊂ I(w j ) for j large enough. Hence, ∩ j j0 I(w j ) = ∅ for some j 0 > 0. Since |w j+1 | > |w j | and W ⊂ V is discrete, we have lim |w j | = ∞ so that lim |I(w j )| = 0, which implies the intersection consists of a single direction and the vectors |w j | −1 w j ∈ I(w j ) converge to it. Thus, θ is well-defined; moreover, θ ∈ I(w j ) for j large enough.
To define Λ(t), we first note by (iii) there exists a j 1 > 0 such that t j < T j for all j > j 1 , while T j < t j+1 for all j 0 since |w j+1 | > |w j |. Let Λ(t) be the continuous piecewise linear function whose graph is broken precisely at the points (T j , M j ) and (t j+1 , m j+1 ) for j j 1 ; it is uniquely determined by requiring its slope be +1 for t T j1 . Hence, each linear piece of Λ(t) has slope For j large enough we have θ ∈ I(w j+1 ) so that Lemma 3.4 implies (18) holds with v = w j+1 . The hypotheses of Lemma 3.1, with w j and w j+1 in place of v and v ′ , are easily verified using (21) and |w j+1 | > |w j |. Hence, we conclude by Lemmas 3.3 and 3.1 that the points (T j , M j ) lie within a uniform bounded distance of the graph of f (t) : It is readily seen from (8) that f ′ is monotone on each I k with absolute value 1. The proof of the proposition is now complete once we show: Claim: the points (t j , m j ) lie within a uniform bounded distance of the graph of f (t).
To prove the claim we shall apply Lemma 3.2 to the vectors w j and w j+1 . The first hypothesis |w j | < |w j+1 | follows by (i). We record here the second and third hypotheses for later reference: Using (21), the triangle inequality and then |w j+1 | > |w j | we have which implies the second hypothesis since g 2. Next, using |w j+1 ×θ| < |w j ×θ| and |w j+1 | √ 2|w j |, which holds by (i) again, we obtain the third hypothesis.
To get an inequality in the other direction, let φ be the angle between g θ t w j and g θ t w j+1 and h the height of the isosceles triangle formed by them. Since w j+1 = w j + gv ′ we have Observe that the distance between any two lines parallel to g θ t v ′ that intersect g θ t Z is an integer multiple of 1/|g θ t v ′ | and the same statement holds if W is replaced by Z. Hence, the length of any vector in g θ t V which is not a multiple of (22) implies sin φ 1/3, i.e. φ is bounded away from 0 and π. Hence, from (23) we see there is a universal constant c > 0 such that ℓ(g θ t V ) ce −m/2 /g. It follows that |m − f (t)| ∈ O(log g) and since g is fixed, this completes the proof of the proposition.

Density of primitive lattice points
The main result of this section is Corollary 4.5. It will be needed in §5 to find, given a vector w ∈ W , vectors w ′ ∈ W such that w ′ = w + gv for some v ∈ Z satisfying certain given inequalities on |v| and |w × v|.

Continued fractions in vector form
Recall each α ∈ R admits an expansion of the form whose terms are uniquely determined except for a two-fold ambiguity when α is rational; e.g. 22/7 = 3 + 1/7 = 3 + 1/(6 + 1/1). The kth convergent of α is the reduced fraction p k /q k that results upon simplifying the expression obtained by truncating (24) so that the last term is a k . The convergents of α satisfy the recurrence relations and the inequalities 1 A rational p/q is said to be a best approximation of the second kind if and this property characterises the convergents of α modulo the 0th convergent a 0 , which is a best approximation to α iff the fractional part of α is 1/2. The following is a useful test for a rational to be a convergent: It will be convenient for us to recast the above facts in vector form. Setting v k := p k , q k , the recurrence relations (25) and the identity (26) and Although (27) can easily be rewritten in vector notation, the resulting expression looks awkward because of the distinguished nature of the coordinate directions. Instead, we shall use the following analog of (27) which is expressed in terms of the vector w := α, 1 and its Euclidean length |w|: To see (32) recall that convergents alternate on both sides of α and (27) follows from the fact that the rational (p k + p k+1 )/(q k + q k+1 ) always lies on the same side of α occupied by p k /q k . (32) follows similarly from a comparison of the components of v k+1 + v k , w and v k+1 in the direction perpendicular to v k . and to all nonzero vectors by Spec(w) = Spec(|w| −1 w). We shall also denote by spec(w) the sequence of Euclidean lengths of vectors in Spec(w).
The next lemma was motivated by (29) and will be needed in §5.
Lemma 4.2. Let w be a vector that makes an angle φ with the y-axis. Then for any v ∈ Z 2 such that |v| cos φ > 1 we have Proof. Let P = P (v, w) be the closed parallelogram that has ±v as two of its vertices, one pair of sides parallel to w and the other pair parallel to the x-axis.
The characterisation of convergents given in (28) is equivalent to the statement that every nonzero u ∈ P ∩Z 2 belongs to the union of the two sides of P parallel to w. Hence, it is enough to verify this statement under the given hypotheses. Apply Lemma 3.3 with θ = |w| −1 w (and V the set of integer lattice points which are not scalar multiples of v) to conclude there is some T for which g θ T v is the shortest vector in V ′ = g θ T (Z 2 − 0). Let E be the inverse image under g θ T of the largest closed disk centered at the origin whose interior is disjoint from V ′ . The boundary of E is an ellipse passing through the points ±v while the interior contains no integer lattice points other than the origin. Without loss of generality, we assume w lies in the first quadrant and v in the upper half plane. There are two cases. First, if v lies to the right of w, then E contains P and we are done. Now, if v lies to the left of w, then let x be the length of a horizontal side of P and y the vertical distance between v and its reflection in the line Rw. It is enough to show x < 1 and y < 1. If z is the distance between v and its reflection then z = 2|v| sin ∠wv = 2|w × v|/|w| 1/|v| < cos φ so that x = z sec φ < 1 and y = z sin φ < 1.
Note if Ω a = {(x, y) : x + y a, x 0, y 0} then dens(2Ω a \ Ω a ) = 2/3a 2 for 1 a < 3/2 and dens(γΩ 1 \ Ω 1 ) = 0 for γ < 2 show that the constants in the preceding lemma are sharp. Let S 1 (Q) be the set of unit vectors of the form v/|v| for some v ∈ Z and S 1 b (Q) the subset formed by those with |v| b. Let I b denote the collection of intervals in S 1 with endpoints in S 1 b (Q). For any interval I ⊂ S 1 let Proof. First we show if I is minimal, i.e. S 1 b (Q) ∩ intI = ∅, then its endpoints correspond to a pair of vectors v, v ′ ∈ Z such that |v × v ′ | = 1. Indeed, there is a linear map γ that sends v to (0, 1) and v ′ to (a, a ′ ) ∈ Z 2 with 0 a ′ < a = |v × v ′ |. If a > 1 then γ −1 (1, 1) ∈ S 1 b (Q) ∩ intI; hence, a = 1. Now observe γΩ(I, b) is a compact convex set satisfying the hypothesis of Lemma 4.3 and since γ preserves density, the proposition holds for minimal I in I b . Since every interval in I b is a (finite) disjoint union of minimal ones, this completes the proof.

Proof.
Let v k ∈ Spec(θ) be the convergent with length |v k | = max spec(θ) ∩ [ε −1 , b]. Then the RHS of (32) implies (the direction of) v k lies in I. Without loss of generality, assume v k lies to the left of θ. Let θ ′ be the right endpoint of I and v ′ l ∈ Spec(θ ′ ) the convergent with length |v ′ l | = max spec(θ) ∩ [1, b]. Note that v k does not lie strictly between v ′ l and v ′ l+1 since the length of any such vector in Z is at least |v ′ l+1 | > b. Nor can v k = v ′ l since the RHS of (32) would imply v ′ l lies strictly to the right of v k . Therefore, v ′ l lies strictly to the right of v k . Let Ω ′ = Ω(J, b) where J is the interval with left endpoint v k and right endpoint v ′ l . The interval I has length |I| = 2 sin −1 (ε/b) πε/b. If |v ′ l | 2ε −1 then the RHS of (32) implies |J| > sin −1 (ε/b) − sin −1 (ε/2b) > ε/2b (we may assume ε < b for otherwise the theorem is easily seen to hold) while if |v ′ l | < 2ε −1 then since any vector strictly between v ′ l and v ′ l+1 has length greater than 2b. In this case, Proposition 4.4 implies dens(2Ω \ Ω) > 4/27π. If |v ′ l | 2ε −1 then arguing as before we see the angle between v k and v ′ l+1 is at least ε/2b > |I|/2π so that we may again conclude that dens(2Ω \ Ω) > 4/27π. Therefore, we may assume v ′ l+1 lies between v k and θ ′ , |v ′ l+1 | 2b and |v ′ l | < 2ε −1 in the remaining. Assuming ε < b/ √ 2 as we may, since the theorem is easily seen to hold otherwise, we obtain the following criterion for a vector left of θ ′ to lie in I: Note that a vector of the form av ′ l + v l−1 lies to the left of θ ′ if a a k+1 . We will show the number of vectors of length b/2 satisfying the above conditions is at least c 0 bε for some absolute constant c 0 > 0. This will complete the proof since between any two vectors of length b/2 (but b) there is a vector whose length is > b and 2b. Using the RHS of (32) we have Assuming |v| b/2 (v = av ′ l + v l−1 ) we see (34) holds for ⌊bε/ √ 2⌋ integers a a l+1 . Among these there are at least b/2|v ′ l | bε/4 which make |v| b/2. Since area(Ω) < πbε/2, we conclude dens(2Ω \ Ω) > 1/6π.

Nonergodic directions and sublinear growth
In this section we prove Theorem 1.
Let e 0 > max(d 0 , 2) be given. The construction involves the choice of a sequence (δ j ) j 0 descending to zero at some prescribed rate as required by Lemmas 5.3 and 5.4 below. For concreteness we set δ j := e 0 j + 1 for all j 0.
In addition, we also fix a constant C := max(2g + 1, c −1 1 e 2e0 ) needed in the statement of Lemma 5.5 below.
Our goal is to find sequences (w j ) j 0 in W satisfying and for all j 0.
Definition 5.1. If w j and w j+1 satisfy (37) and (38) then we say w j+1 is a child of w j . More precisely, (37) and (38) define a family {≺ j } of binary relations on W and to say w j+1 is a child of w j is equivalent to the statement w j ≺ j w j+1 .
Definition 5.2. We say (w j ) j 0 is admissible if |w 0 | > 1, w 0 ± (x 0 , y 0 ) ∈ gZ 2 and for all j 0, w j+1 is a child of w j . A finite sequence (w 0 , . . . , w k ) is admissible if the latter condition holds for 0 j < k.
The choice of (δ j ) was motivated by the next two lemmas, which are stated more generally for a sequence of positive δ j .
Lemma 5.3. If (δ j ) j 0 is a sequence of positive real numbers such that lim inf jδ j > 1 and e 0 = lim sup jδ j < ∞ and (w j ) j 0 is an admissible sequence then lim |w j | −1 w j is a slowly divergent nonergodic direction whose sublinear rate is at most 1 − 1/e 0 .
Proof. First note that the hypotheses imply (i) lim δ j = 0 and (ii) x. Using (i) we may fix K > 1 so that R j S j KR j for all j. Then lim R j = ∞ by (ii). Let c > 1 be given. Using (i) again we fix j 0 large enough so that δ j √ c log(1 + δ j ) for all j j 0 . Now S j KR j0 + √ cR j cR j for all large enough j. Since c > 1 was arbitrary, this poves the claim.
From the first inequality in (37) we have log |w j | (log |w 0 |) i<j (1 + δ i ). Now lim inf jδ j > 1 implies for some p > 1 and C ′ > 0 we have S j p log j − C ′ for all j > 0. The same statement for R j holds by the preceding claim. Hence, c 1 j p for some c 1 > 0. The last two inequalities in (37) imply the cross products form a summable series while (38) implies |w j | is increasing. Hence, lim |w j | −1 w j is a nonergodic direction, by Lemma 2.4.
It is clear from the preceding that lim |w j | = ∞; however, a much stronger statement holds. First, there is some p > 1 and c 2 > 0 such that (for all j > 0) Now the second inequality in (37) implies log |w j | (log |w 0 |) i<j (1 + δ i ) + j log C. Using lim sup jδ j < ∞ and arguing as before we find q > e 0 and C ′′ > 0 such that R j q log j + C ′′ for all j > 0. Thus i<j (1 + δ i ) c 3 j q some c 3 > 1 and log |w j | c 3 j q log |w 0 |+j log C. Using log(1+x+y) log(1+x)+log(1+y) we conclude: for some q p and C ′′′ > 0 we have (for all j > 0) log log |w j | log log |w 0 | + (q + 1) log j + C ′′′ .
It follows from (39) and (40) that lim |w j | δj / log |w j | = ∞. The hypotheses of Proposition 3.6 are satisfied since (i) is the same as (38) while (37) and (39) imply lim |w j |/|w j+1 | = 0 and lim |w j × w j+1 | = 0, where the ratio of consecutive cross-product terms is bounded. Propositions 2.1 and 3.6 imply lim |w j | −1 w j is slowly divergent since while the sublinear rate is at most . The proof is completed by observing that q > e 0 may be chosen arbitrarily close to e 0 .
Lemma 5.4. Let (δ j ) j 0 be a sequence of positive real numbers such that lim inf jδ j > 2 and lim sup jδ j < ∞ and (w j ) j 0 an admissible sequence. Then for any ε > 0 there exists Proof. Repeating the arguments in the preceding proof with the stronger hypotheses we find there are constants p > 2, c 2 > 0, q p and C ′′′ > 0 such that (39) and (40) hold for all j > 0. It follows that the difference for some c ′ 2 > 0. The function β(j) is increasing for j 1 and by choosing L 0 large enough we have β(1) − log ε. By choosing L 0 even larger so that (log |w 0 |) 2 /|w 0 | δ0δ1 ε we obtain (41).
Proof. Apply Corollary 4.5 with ε −1 = e t |w j | log |w j | and b = |w j | 1+δj to get ρ 1 e −t |w j | δj / log |w j | vectors v ∈ Z satisfying the inequalities The vector w j+1 = w j + gv satisfies (38) by the first inequality in (44), which together with the second inequality implies |w j+1 | |v| |w j | 1+δj . The third implies |w j+1 | |w j |+g|v| (2g+1)|w j | 1+δj , which together with the remaining inequalities and |w j × w j+1 | = g|w j × v| implies (37) for the given value of C. Therefore, w j+1 is a child of w j and since t 2e 0 , this proves the first part.
Using |w j+1 | > g|v|, the last inequality in (44) and g 2 we have as soon as log |w 0 | > 1. Since v ∈ Z, Lemma 4.2 implies +v is a convergent of w j+1 , where the sign follows from w j+1 · v > 0 and the fact that all angle between a vector and its convergents are acute. (See Remark 5.6 below for an explanation of how the hypothesis of Lemma 4.2 is satisfied.) Let v ′ ∈ Spec(w j+1 ) be the next convergent after v. Using the RHS of (32), the second to last inequality in (44), t 2e 0 , |w j | < |w j+1 | and (41) we have provided |w 0 | is chosen large enough as required by Lemma 5.4 for ε = c 1 e −2e0 .
Remark 5.6. In order to satisfy the hypothesis of Lemma 4.2 one needs to make a minor technical assumption that the angles φ j made between the vectors w j of an admissible sequence and the y-axis are bounded away from π/2. This can be ensured by choosing φ 0 close to the y-axis, using (37) and the cross product formula to control the angles ∠w j w j+1 , and then requiring |w 0 | large enough.
It would be desirable if the conclusion of Lemma 5.5 could be strengthened so that the newly constructed vectors satisfy (43) without the "−δ j " in the exponent, for then we can use the lemma to construct admissible sequences by recursive definition. However, it can be shown that this stronger statement is false. (This uses a result of Boshernitzan-see the appendix to [Ch].) Fortunately, the induction can be rescued by using a slight variation of the condition (42).
Let W j be the set of w ∈ W with the following property: for all t δ j , spec(w) ∩ [e t |w| log |w|, |w| 1+t ] = ∅.
Proof. If (45) does not hold for some t e 0 , then w has convergents v k and v k+1 satisfying |v k | < e t |w| log |w| and |v k+1 | > |w| 1+t . On the one hand we have |w × v k | < |w|/|v k+1 | < |w| −t by (32); on the other hand we have (3) implies Therefore, if L 0 is chosen large enough so that the RHS is < e 0 for |w| L 0 , then (45) holds for all t e 0 .
Proposition 5.8. There exist L 0 > 0 and ρ 2 > 0 such that if w j ∈ W j belongs to an admissible sequence (w 0 , . . . , w j ) with |w 0 | L 0 , then it has ρ 2 |w j | δj / log |w j | children contained in the set W j+1 .
Proof. Let v k ∈ Spec(w j ) be the unique convergent of w j determined by the condition |v k | |w j | 1+δj < |v k+1 | so that |v k | = e t1 |w j | log |w j | and |v k+1 | = |w j | 1+t2 for some t 1 δ j and t 2 t 1 . If t 1 e 0 + δ j then w j satisfies the hypothesis of Lemma 5.5 with t = e 0 + δ j and each child constructed by the lemma satisfies (43), which is easily seen to imply (45) for t ∈ [δ j+1 , e 0 ]. By Lemma 5.7 it follows that all children constructed lie in the set W j+1 . Therefore, the conclusion of the proposition holds in this case for ρ 2 = ρ 1 e −2e0 . Now consider the case t 1 < e 0 + δ j . This time Lemma 5.5 is applied with t = t 1 to obtain the same number of children as before, each satisfying (43) with t replaced by t 1 . Let W ′ consist of those children which do not belong to W j+1 . Our goal is to show W ′ occupies only a small fraction (independent of j) of all the children constructed, provided L 0 is large enough.
Let ϕ : W ′ ֒→ Z be the function that assigns to any w j+1 ∈ W ′ the unique convergent v ′′ ∈ Spec(w j+1 ) with maximal Euclidean length |v ′′ | |w j+1 | 1+δj . The plan is to show ϕ is injective then control the cardinality of its image.
First, we claim every v ′′ ∈ im ϕ satisfies an inequality of the form for some positive integer a < e e0 and a choice of the sign on the LHS. Indeed, suppose v ′′ = ϕ(w j+1 ) and let v ′′′ be the next convergent after v ′′ . Then |v ′′ | = e t3 |w j+1 | log |w j+1 | and |v ′′′ | = |w j+1 | 1+t4 for some real numbers t 3 t 1 − δ j , since w j+1 satisfies (43), and t 4 > δ j+1 , by definition of v ′′′ . Note that t 4 > t 3 because 2 w j+1 ∈ W ′ . By the RHS of (32) and the first inequality in (37) we have Recalling the consecutive pair of convergents v and v ′ constructed in the proof of Lemma 5.5 we see that v ′′ = av ′ + bv for some positive integers a and b, except in the case when v ′′ = v ′ . In any case a > 0 and the definition of v ′′ implies these are the only possibilities. Recall also that v ′ is the convergent responsible for (43) and since t 1 δ j it follows that |v ′ | |w j+1 | log |w j+1 |. On the other hand, we have t 3 < e 0 , for otherwise Lemma 5.7 would imply w j+1 ∈ W ′ ; therefore |v ′′ | < e e0 |w j+1 | log |w j+1 | and since |v ′′ | > a|v ′ | (because angles between convergents are acute) we have a < e e0 . Finally, observe that by (31). This proves the claim.
Next, we show ϕ is injective. Let v ′′ i = ϕ(w i j+1 ) for i = 1, 2 and recall in the proof of Lemma 5.5 it was shown that w i j+1 = w j +gv i for some v i ∈ Spec(w i j+1 ). Obviously, v 1 = v 2 since we are assuming w 1 j+1 = w 2 j+1 . Using (44), we see that which is approximately a factor of log |w j | greater than the angle either vector makes with the corresponding child: Since the angle between v ′′ i and w i j+1 is even smaller than the above, it follows that the vectors v ′′ 1 and v ′′ 2 are also distinct. Finally, we bound the number of vectors in im ϕ. Let v ′′ i for i = 1, 2 be two vectors in im ϕ satisfying (46) with the same sign and the same positive integer a. Put u = v ′′ 1 − v ′′ 2 and recall the definition of v k ∈ Spec(w j ) at the beginning of the proof. Claim: If |u| (1/4)|w j | 1+δj /4 then u = ±dv k for some positive integer d < 4|w j | δj (1−δj+1) . The claim implies we either have Observing that the former is much greater than the latter, which means according to (46) the vectors in W ′ are contained in < 2e e0 narrow strips parallel to w j and within each strip there are < 2(2g + 1)e 2e0 log |w j | clusters 3 each having < 4|w j | δj (1−δj+1) vectors. If L 0 is chosen large enough as required by Lemma 5.4 for ε = 32 −1 (2g + 1) −1 e −2e0 ρ 1 , then it follows that less than half the children constructed lie in W ′ , i.e. assuming the claim, the proposition holds in this case with ρ 2 = (1/2)ρ 1 e −2e0 .
Therefore, v = ±v k ′ for some convergent v k ′ ∈ Spec(w j ). (See the Remark 5.6 for an explanation of how the hypothesis of Lemma 33 is satisfied.) Since |v| < 3 A rough estimate: |u| 2e t 3 |w j+1 | log |w j+1 | < 2(2g + 1)e 2e 0 |w j | 1+δ j log |w j | provided |w 0 | is large enough. Here, we used w i j+1 ∈ W ′ and Lemma 5.7 to get |v ′′ i | < e e 0 |w j+1 | log |w j+1 | then apply |w i j+1 | < (2g + 1)|w j | 1+δ j . |w j | 1+δj by hypothesis, k ′ k by definition of v k . In fact, we must have equality for if k ′ < k then using the LHS of (32), the fact that Euclidean lengths of convergents form an increasing sequence, and t 1 2e 0 , we have (47) if L 0 is chosen large enough as required by Lemma 5.4 for ε = (1/4)e −2e0 . Using (47) and the preceding facts about continued fractions once again, we get This proves the claim, and hence the proposition.
Proof of Theorem 1. Since δ 0 = e 0 Lemma 5.7 implies any vector in W with large enough Euclidean length belongs to W 0 . Choose any w 0 ∈ W 0 with |w 0 | greater than the value of L 0 given by Proposition 5.8. (In addition, require that the initial direction be chosen as in Remark 5.6.) Applying Proposition 5.8 inductively we construct an infinite number of admissible sequences (w j ) with the property w j ∈ W j for all j 0. Moreover, Lemma 5.3 implies the directions of the vectors in each sequence converge to a slowly divergent nonergodic direction with sublinear rate 1 − 1/e 0 . Any vector w j occuring at the jth stage of the construction has at least m j = ρ 2 |w j | δj / log |w j | children for which the inductive process may be continued indefinitely. The angle between the directions of these children are at least where c > 0 is constant depending only on g. Using [Fa,Example 4.6], we see the Hausdorff dimension of the set of directions constructed is at least

Arbitrarily slowly divergent directions
To prove Theorem 2 we shall show given any function R(t) with R(t) → ∞ as t → ∞ there exists a sequence (w j ) satisfying the hypotheses of Propostion 3.6 together with M j R(T j ) for j large enough and lim m j = ∞. Note that the length of the shortest simple closed curve on (X θ t , q θ t ) is at most 2ℓ(g θ t V ) since it consists of at most two saddle connections, both corresponding to the same vector in g θ t V . Therefore, lim m j = ∞ implies lim |w j | −1 w j is divergent. The notation A ≍ B means A/C B AC for some implicit universal constant C > 0. Also, A ≪ B means A Bε for some implicit constant ε > 0 that may be chosen as small as desired at the beginning of the construction. B ≫ A is equivalent to A ≪ B.
Proposition 6.1. If r(t) increases to infinity and r(t)r ′ (t) → 0 as t → ∞, then there is a sequence (w j ) j 0 satisfying the hypotheses of Proposition 3.6 such that for all j large enough (a) m j r(t j ) + C, where C := log 2, Proof of Theorem 2 assuming Proposition 6.1. Observe the hypotheses of the proposition is satisfied by the logarithm of any smooth Lipschitz function increasing to infinity. Therefore, given R(t) and any ε > 0 one can readily find r(t) satisfying the hypotheses and R(t) (2 + ε)r(t).
Here, we used the fact that vectors in W have irrational slope. The next lemma estimates the length of u.
Lemma 6.2. If |w| > b|v| and |w × v| ε 1/2 √ 2 then Proof. First, consider the case where w lies between u and v. Then u + v lies between w and v so that comparing the component of the vectors u, w and u + v orthogonal to v we obtain Let a, b ′ , c > 0 be given by |v| = a|u|, |v| = b ′ |w| and c = |w × v|. Using |u + v| (1 + a)|u| we note the LHS implies |w| |u + v|c (1 + a)c|u| But then |v| (1 + a)b ′ c|u| so that |u + v| (1 + (1 + a)b ′ c)|u|. Repeating the above argument starting with new estimate on |u + v| we get |u + v| (1 + (1 + (1 + a)b ′ c))b ′ c|u|. By induction, we find |u + v| |u|/(1 − b ′ c). Since bb ′ < 1 this gives the LHS of (48) while the RHS holds trivially. Now consider the case where u lies between w and v. Then w lies between u and u − v so that comparing the component of u w and u − v orthogonal to v we get 1 |u| |w × v| |w| Using |u − v| (1 − a)|u| we note the RHS implies  (1 − a)b ′ c))b ′ c|u|. By induction, we find |u − v| |u|/(1 + b ′ c). Since bb ′ < 1 this gives the RHS of (48) while the LHS holds trivially.
There are no other cases because Lemma 3.4 implies |u| > |v| so that v does not lie between w and u.
Proof of Proposition 6.1. Let w 0 ∈ W be arbitrary. Since w 0 has irrational slope, we may choose w 1 = w 0 + gv 1 for some v 1 ∈ Z so that |w 0 × w 1 | = g|w 0 × v 1 | ≪ 1, and in particular, 1/2 √ 2. Since r(t) is slowly increasing, the choice can be made so that m 1 ≫ r(t 1 ).
Given (w j , v j ) ∈ W × Z with |w j × v j | < 1/2 √ 2 let u j be the unique vector in Z satisfying |w j × u j | < 1 2 |w j × v j |, |u j × v j | = 1, and w j · u j > 0 (51) and define v i j+1 , i = 0, 1, 2 by v 1 j+1 = u j + σv j , v 2 j+1 = 2u j + σv j and v 0 j+1 = u j − σv j where σ = +1 if w j lies between u j and v j and −1 otherwise. We note here and |w j × v 2 j+1 | |w j × v 1 j+1 |. The next pair (w j+1 , v j+1 ) will be chosen among the three possibilities (w i j+1 , v i j+1 ) where w i j+1 = w j + gv i j+1 . Note that u j and v j are uniquely determined by w j . Hence, we may let δ = δ(w j ) ∈ (0, 1/2) be defined by The index i ∈ {0, 1, 2} is determined according to the following rule: Similarly, for either choice i = 1, 2, we have m j+1 m j . If m j r(t j ) + C for j = j 0 then the same holds for all j j 0 . Since r(t) increases to infinity and m j+1 m j whenever (A) is used to choose the next vector, m j r(t j ) + C for some j. This proves (a).
Since (A) is used to choose the next vector for at most finitely many j, from some point on the only situation when m j+1 < m j is if i = 0 in (B) is used to choose the next vector, but then |m j+1 − r(t j+1 )| C. This proves (b).
By choosing v 1 so that r(t 1 ) ≫ 1 we can ensure that m j ≫ 1 for all j 1, since r(t) is increasing. In other words, for any ε > 0 we can choose v 1 so that the sequence of pairs (w j , v j ) constructed satisfy |w j × v j | ε. We have |w j | ∈ O(ε|u j |) by Lemma 6.2 so that |v j+1 | ≍ |u j | ≍ |w j |e mj for ε small enough. This proves (d) and using |v j+1 | ≫ |w j | it is readily verified that (w j ) satisfies the hypotheses of Proposition 3.6.
It remains to prove (c). The hypothesis implies (C) is used to choose the next vector with either i = 1, 2. In this case, m j+1 = m j + log 1/(1 − iδ(w j )) > m j + δ(w j ).