Veech Surface Research Articles

A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle

We establish a geometric criterion on a $SL(2, R)$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $SL(2, R)$-orbits of all algebraically primitive Veech surfaces (see also [7]) and of all Prym eigenforms discovered in [34], as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also [4, 17]). The argument simplifies and generalizes our proof for the case of canonical measures [17]. In the Appendix, Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.

Modular Fibers and Illumination Problems

For a Veech surface (X, !), we characterize Aff + (X, !) invariant subspaces of X n and prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very par- ticular shape (in any dimension). Among other consequences we find copies of (X, !) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X, !) prelattice we prove the at most countableness of points non-illuminable from any x ∈ X. Applying our results on invari- ant subspaces we prove the finiteness of these sets when (X, !) is Veech.

Veech surfaces with nonperiodic directions in the trace field

Veech's original examples of translation surfaces $\mathcal V_q$ with what McMullen has dubbed optimal dynamics'' arise from appropriately gluing sides of two copies of the regular $q$-gon, with $q \ge 3$. We show that every $\mathcal V_q$ whose trace field is of degree greater than 2 has nonperiodic directions of vanishing SAF-invariant. (Calta-Smillie have shown that under appropriate normalization, the set of slopes of directions where this invariant vanishes agrees with the trace field.) Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant. In an appendix, we prove various elementary results on the inclusion of trigonometric fields.

Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main tool is the Hodge-theoretic characterization of Teichmüller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmüller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmüller curve.

Minimal non-ergodic directions on genus-2 translation surfaces

It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for any genus-2 translation surface which is not a Veech surface there are uncountably many minimal but not uniquely ergodic directions. The theorem can be applied to certain billiard tables.

Unipotent flows on the space of branched covers of Veech surfaces

There is a natural action of SL $(2,\mathbb{R})$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \big\{\big(\begin{smallmatrix}1 & * 0 & 1\end{smallmatrix}\big)\big\}$ . We classify the U -invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL $(2,\mathbb{R})$ -orbit of the set of branched covers of a fixed Veech surface.) For the U -action on these submanifolds, this is an analogue of Ratner's theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is $2 \pi/n$ , with $n \ge 5$ and n odd.

Veech surfaces and complete periodicity in genus two

We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair ( M , ω ) (M, \omega ) where M M is a Riemann surface and ω \omega is an Abelian differential on M M . Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of ω \omega . A direction v v on a translation surface is completely periodic if any trajectory in the direction v v is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction v v . Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group S L ( 2 , R ) SL(2, \mathbb {R}) on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false. In this paper, we use the J J -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space H ( 2 ) {\mathcal H}(2) of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.