Comptes Rendus Mathématique

. The initial portion of the Lagrange spectrum L B 7 of certain square-tiled surfaces of genus two was described in details in the work of Hubert–Lelièvre–Marchese–Ulcigrai. In particular, they proved that the smallest element of L B 7 is an isolated point φ 1 , but the second smallest value φ 2 of L B 7 is an accumulation point. Also, they conjectured that the portion L B 7 ∩ [ φ 2 , η 1 ) is a Cantor set for a speciﬁc value η 1 and they asked about the continuity properties of the Hausdor ﬀ dimension of L B 7 ∩ ( −∞ , t ) as a function of t < η 1 . In this note, we solve a ﬃ rmatively these problems.


Introduction
The classical Lagrange spectrum L was originally introduced in relation to the study of Diophantine approximations of irrational numbers and, alternatively, it can also be seen as the set of real numbers encoding cusp excursions of geodesics on the modular surface, i.e., L = lim sup t →∞ 2 sys(g t (X )) 2 < ∞ : X ∈ SL(2, R)/ SL(2, Z) , where g t := diag(e t , e −t ) and sys(Y This point of view led Hubert-Marchese-Ulcigrai [5] to naturally extend the notion of Lagrange spectrum to the context of Teichmüller dynamics (see, e.g., Zorich's survey [8] for the basic aspects of this theory).
More concretely, they defined the Lagrange spectrum L I associated to the closure I of a SL(2, R)-orbit on the moduli space of unit area translation surfaces as where the action of g t is the so-called Teichmüller geodesic flow and sys(Y ) is the minimal length of a saddle-connection of Y .Also, they showed that L I shares some common features with the classical Lagrange spectrum, e.g., • if I consists of some translation surfaces with genus g and σ conical singularities, then L I is a subset of , ∞ given by the closure of the maximal values of the function Y → 2 sys(Y ) 2 along g t -periodic orbits included in I ; • if I contains a square-tiled surface, then L I contains a Hall's ray, i.e., [r, ∞) ⊂ L I for some r > 0.
On the other hand, it was discovered by Hubert-Lelièvre-Marchese-Ulcigrai [4] that the beginning of the Lagrange spectra of SL(2, R)-orbits of square-tiled surfaces might behave differently from the classical Lagrange spectrum.More precisely, let X be the square-tiled surface of genus two with unit area obtained from seven squares sq(k), 1 ≤ k ≤ 7, in R 2 with areas 1/7 by gluing the right vertical side of sq(k) to the left vertical side of sq(h(k)) and the top horizontal side of sq(k) to the bottom horizontal side of sq(v(k)), where h and v are the permutations with cycles h = (1, 2, 3)(4)(5)(6) (7) and v = (1, 4, 5, 6, 7)(2)(3).
For the sake of exposition, we divide the rest of this note into five sections: first, we review some results from [4] about the description of the initial portion of L B 7 ; next, we employ the results of Cerqueira, Moreira and the author [2] to deduce the continuity of the Hausdorff dimension of L B 7 ∩ (−∞, t ) as a function of t ∈ (−∞, η 1 ); afterwards, we show that K is a Cantor set; then, we modify an argument of Moreira [7] in order to prove that any φ ∈ K is accumulated by Cantor sets with positive Hausdorff dimensions contained in K; finally, we show that d (t ) is not Hölder continuous near φ 2 .

Preliminaries
Consider the left shift dynamics σ : {a, b} Z → {a, b} Z on the symbolic space Σ := {a, b} Z where a := 1, 4, 2, 4 and b := 1, 3. It was shown in [4, §4.5] that where k n is an explicit increasing sequence converging to 3. 2 We are using the notations [a 0 ; a 1 , . . .] = a 0 + where h : Σ → R is the height function given by (i) there exists k ≥ 2 such that where c ( j ) := c . . .c j times , or (ii) there are k, n ≥ 1 such that φ = L σ (ξ) where ξ ∈ Σ contains infinitely many copies of a (k) b (n) a but no copies of a (k+1) and no copies of a (k) b (n−1) a.Since Theorem 1.1 of their article ensures that φ 2 , φ ∞ ∈ K , we have that K is a closed set without no isolated points.

Proof of Theorem 1
It is well-known [1] that the left-shift dynamics on {1, 2, 3, 4} Z can be smoothly realized via the natural extension ϕ(x, y) = ({1/x}, 1/( 1/x + y)) of the Gauss map g ([0; a 1 , a 2 , . . .]) := [0; a 2 , . . .]. Since ϕ is a smooth area-preserving diffeomorphism whose local stable and unstable manifolds are parallel to the axes and the gradient of the smooth realization of the height function h is transverse to the axes, the key results from [2] can be employed to derive that: , where D(η 1 ) is the Hausdorff dimension of Cantor set C (a, b) of real numbers with continued fraction expansions in Σ + = {a, b} N .
At this point, the desired theorem follows from the fact that L

Proof of Theorem 2
We saw in Section 2 that K is a perfect set.Therefore, our task of showing that K is a Cantor set can be reduced to prove that d (η 1 ) = 2 • D(η 1 ) < 1.
In the sequel, we will show that D(η 1 ) = 0.154 . . . .For this sake, we observe that where and ψ : . Hence, we can use the method described in [6, §4]

Local structure of K
Recall that K is a Cantor set.In particular, any x ∈ K is accumulated by a sequence x n ∈ K with In what follows, we adapt the proof of Theorem 3 in [7] to show that x is accumulated by Cantor sets of positive Hausdorff dimensions included in K.
In this direction, let us take ξ and, for each j and n as above, consider the finite sequence with 2N + 1 terms (ξ (n) j −N , . . ., ξ (n) j , . . ., ξ (n) j +N ) =: S( j , n).By the pigeonhole principle, there exists a finite string S such that, for infinitely many values of n, the string S appears infinitely many times as S( j , n), i.e., there is an infinite set A ⊂ N so that for each n ∈ A we can find j 1 (n) < j 2 (n) < . . .with lim i →∞ ( j i +1 (n) − j i (n)) = ∞ and S( j i (n), n) = S for all i ≥ 1.
to obtain that, for all n ∈ N,