ON HAUSDORFF DIMENSION OF THE SET OF NON-ERGODIC DIRECTIONS OF TWO-GENUS DOUBLE COVER OF TORI

. Cheung, Hubert and Masur [Invent. Math., 183(2011), no.2, pp. 337-383] proved that the Hausdorﬀ dimension of the set of nonergodic directions of billiards in a kind of rectangle with barrier is either 0 or 12 . As an application of their argument, we prove that there exist the third-kind two-genus double covers of tori in which the set of minimal and non-ergodic directions have Hausdorﬀ dimension 12 .


Introduction.
A translation surface has a collection of directional flows which preserves the Lebesgue measure induced by its translation structure. It is classical that the subset of ergodic directions is of full measure. See [7].
For quantifying the subset of non-ergodic directions further, it is naturally to consider the Hausdorff dimension. In fact, the Hausdorff dimension provides a functor on the moduli space of translation surfaces, which is invariant under the action of G =GL + (2, R) and is constant almost everywhere with respect to its natural measure. See [8].
Non-ergodic directions in two-genus translation surfaces are studied extensively. In [4], Cheung and Masur proved that for a two-genus translation surface either it is a Veech surface, or it has uncountable non-ergodic directions. On the other hand, the Hausdorff dimension of the set of non-ergodic directions is equal to 1 2 for almost every translation surface in the stratum H (2). See [1].
For any λ ∈ (0, 1) let Q λ be a 1 2 -by-1 rectangle with a horizontal barrier of length 1−λ 2 initiated from the middle point of the vertical side. There is a standard procedure to turn billiards in Q λ into the directional flows on a translation surface. Denote the translation surface by X λ . It is easy to see that X λ be the resulting surface by gluing two copies of T 2 together along a horizontal geodesic segment of length λ. See [2], [3] and [5]. It seems that by now X λ is the only translation surface, of which the Hausdorff dimension of the set of non-ergodic directions is calculated concretely, except Veech surfaces.
Let E be the space of two-genus translation surfaces, which is a double cover of a flat torus. Then E ⊂ H(1, 1) is an affine invariant sub-manifold of complex dimension 3. Currently, the study of affine invariant sub-manifold is active. See [10]. It should be interesting to consider the Hausdorff dimension functor restricting on affine invariant sub-manifolds.
A saddle connection γ of a two-genus translation surface is said to be separating if the complement of γ ∪ j(γ) has two connected components, where j is the hyperelliptic involution. The corresponding direction is also said to be separating. It is known that a two-genus translation surface always has infinitely many separating directions. See [10] and [9]. Recall that a direction in a translation surface is periodic if there is a closed geodesic in this direction. It is obviously that the classification of surfaces in E is invariant under the Gaction. Moreover, X ∈ E is of the first kind if and only if X is a Veech surface, which means that its non-ergodic directions are countable; the second kind if and only if X belongs to the G-orbit of X λ for a unique irrational λ. In 2011 Cheung, Hubert and Masur [3] proved that the set of nonergodic directions on X λ has Hausdorff dimension either 1 2 or 0, depending on the sequence of k-th convergent of λ satisfies the Pérez-Marco condition log log q k+1 q k < ∞ or not. Hence it would be desirable to know whether the set of non-ergodic directions on any double cover X ∈ E has Hausdorff dimension either 1 2 or 0. In fact, we get the following result: There exist X ∈ E of the third kind such that the Hausdorff dimension of the set of its minimal and non-ergodic directions is equal to 1 2 . The paper is organized as following: In Section 2 one classifies two-genus double covers of tori; Theorem 3.1 is proved in Section 3 under the existence of good trees of slits; Section 4 is devoted to prove the existence of such trees.
2. Double cover of tori. To generalize X λ , denote by X φ , 0 ≤ φ, φ < 1, the resulting surface of gluing two copies of the flat torus T 2 along an oriented geodesic segment with two different end points, of which the holonomy is equal to φ. See Figure 1. Suppose that X ∈ E. Then X belongs to the G-orbit of X φ for some φ.
Proof. The famous Riemann-Hurwitz's formula implies that E ⊂ H(1, 1). Since the moduli space of one-genus translation surfaces H 1 = G/SL(2, Z), there exists Y in the G-orbit of X such that Y is a double cover of the flat torus T 2 . Without loss of generality, assume that the images of two singular points of Y are [0] and [φ]. As shown in Figure 2, it turns out that either

Figure 2. Combinatorial realization
Corollary 1. Let X ∈ E be a double cover. Then X is of the first kind if and only it is Veech surface; X is of the second kind if and only its G-orbit contain X λ with λ / ∈ Q, and such λ is unique.
A slit of X φ is defined to be the holonomy of an oriented separating saddle connection in X φ . Denote by S(φ) the set of all slits of X φ .
Proof. Cut the flat torus T 2 with two marked points [0] and [φ] along three saddle connections with holonomies φ, 1 + i − φ and 1 − φ. one gets a hexagon. Because X φ is a double cover of T 2 branched at [0] and [φ]. X φ can be realized as two copies of the hexagons gluing together along the side.
Cutting the two hexagons along the geodesic line determined by φ + m + in results a finite number of pieces of polygon. To get X φ+m+in , at first glue these polygons along the sides determined by φ + m + in to get two new hexagons; next glue these new hexagon along sides determined by φ, 1+i−φ and 1−φ. See Figure 3. Then some tedious manipulation yields that X φ+m+in = X φ if and only m + in ∈ 2Z[i].
Notation. For any w ∈ S(φ) denoted by (w) the real part of w and (w) its imagine part; denote by |w|, named the height of w, the absolute value of (w); and denote by α w , named the inverse slope of w, the quotient (w) (w) .

YAN HUANG
3. Proof of Theorem 1.2. Let NE(φ) be the set of inverse slopes of minimal and non-ergodic directions in X φ and let H.dim(φ) be the Hausdorff dimension of NE(φ). Recall that an irrational number α is Diophantine if there exist constants c > 0 and e > 0 such that |α − p q | ≥ c q e for any integers p and q > 0.
if otherwise, X φ is of the third kind. Hence Theorem 3.1 implies the main theorem.
Proof. Since it follows from (1) that {α wj } is a Cauchy sequence.
Proposition 4. Suppose that w and w are two different δ-children of w ∈ S(φ).
On the other hand, As a result, for sufficiently large |w|. Definition 3.3. For any δ > 0, a δ-tree V of slits is a disjoint union of subsets V j , j ≥ 0, of S(φ) such that there is a unique 0-level slit and each (j + 1)-level slit is a δ-child of some j-level slit. And V is good if there exists a constant C such that each slit w ∈ V has at least C|w| δ / log |w| δ-children in V .
For any w ∈ S(φ) denote by I(w) the closed interval centered at α w with diameter 4 |w| 2+δ . For any δ-tree V = ∪V j of slits denote K(V ) = j K(V j ), where K(V j ) = w∈Vj I(w).
Assume that w ∈ V j is a slit of level j and w and w are two δ-children in V j+1 of w. Since there is a unique 0-level slit w 0 in V and each (j + 1)-level slit is a δ-child of some j-level slit, it turns out that K(V j ) is a disjoint union of finitely many closed intervals such that K(V j+1 ) ⊂ K(V j ) and provided that 5 1 δ 2 ≤ |w 0 |. Then the distance dist(I(w ), I(w )) between I(w ) and I(w ) satisfies that If V is good, then there is a constant C such that I(w) contains at least (k + 1) log(1 + δ) + j log(C log |w 0 |)

4.
Existence of good tree. The section is devoted to the existence of good δ-tree, of which the unique 0-level slit has arbitrarily large height. For any irrational number α denote by Spec(α) the sequence formed by heights of the convergents of α. Let p k q k be the k-th convergent of α. It is well-known that 1 and where a k is the k-th partial quotient of α. Conversely, if p and q > 0 are integers satisfying then p q is a convergent of α, where p q is unnecessarily reduced. The hypothesis that (ηφ) is Diophantine implies that there are constants c 0 > 0 and e 0 > 0 such that for any w ∈ S(φ), p + iq ∈ Z[i] with |q| > 0 and n ∈ Z. Fix a real number N such that e 0 < N δ.
Proof. By Proposition 6, it suffices to prove the existence of N -good slits with arbitrarily large height.
LetṼ be the collection of nice children of w, which are not normal. It is obviously thatṼ = ∅ when n 1 ≥ N + 1. It remains to consider the case that n 1 < N + 1. Proof. Since n 1 < N + 1, it follows that n = n 1 . The definition of q l implies that q l +1 = |w | 1+n δ for some real number n > 1. Since w is (n − 1)-good, it follows that q l = e n δ |w | log |w | for some real number n ≥ n 1 − 1 ≥ 0. Using Proposition 6, w is not N -good so that n < N ; this proves (i).
Hence a ≤ e N δ .