Diophantine approximation on Veech surfaces

We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments

Circle averages and disjointness in typical flat surfaces on every Teichmueller disc

We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with this property. We provide an application to the convergence of `circle averages' for the flow (away from a sequence of radii of density 0) for such surfaces. Even the density of a sequence of 'circles' was only known in a few special examples.Comment: 14 page

Veech groups without parabolic elements

We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.Comment: 7 pages, Corrected typos, to appear in Duke Mathematical Journa

Prime arithmetic Teichmuller discs in H(2)

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two all translation surfaces in H(2) tiled by a prime number n > 3 of squares fall into exactly two Teichmuller discs, only one of them with elliptic points, and that the genus of these discs has a cubic growth rate in n.Comment: Accepted for publication in Israel Journal of Mathematics. A previous version circulated with the title "Square-tiled surfaces in H(2)''. Changes from v1: improved redaction, fixed typos, added reference

On the Hausdorff dimension of the Rauzy gasket

In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is less than 2. By this result, we answer a question addressed by Pierre Arnoux. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Ya. Maltsev in 2003.Comment: 23 pages, 5 figure