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I am reading this paper of Avila and Delecroix of the billiard flow on polygonal surfaces, but I have to get through some basic definitions first. What is a non-arithmetic Veech surface?

A Veech surface is an “exceptionally symmetric” translation surface, in the sense that the Veech group is a (finite co-volume) lattice in $SL(2, \mathbb{R})$ (it is easily seen that the Veech group is never co-compact). Simple examples of Veech surfaces are square-tiled surfaces, obtained by gluing finitely many copies of the unit square $[0, 1]^2$ along their sides: in this case the Veech group is commensurable with $SL(2, \mathbb{Z})$.

These are the kinds of surface they want to rule out (because they will not lead to weak-mixing flows. The promise is, that as long as you're not scared of decimal error, these problems have an elementary flavor, can be simulated on a computer. The proofs might not be.

Veech surfaces that can be derived from square-tiled ones by an affine diffeomorphism are called arithmetic. Arithmetic Veech surfaces are branched covers of flat tori, so their directional flows are never topologically weak mixing (they admit a continuous almost periodic factor).

Their example of a non-arithmetic Veech surface is the regular polygon $P_n$ (with opposite sides identified) and or a copy of $P_n$ and its mirror $-P_n$ if $n = 2k+1$. $S_n = P_n$ or $S_n = P_n \cup (- P_n)$. Then they show two results. The first one is slightly more refined than the Veech dichotomy because it says these directions are weak-mixing.

Thm 2 The geodesic flow in a non-arithmetic Veech surface is weakly mixing in almost every direction. Indeed the Hausdorff dimension of the set of exceptional directions is less than one.

And we get rare examples of weak-mixing dynamical systems (that are not mixing). The elements of the Veech group are $2\times 2$ matrices living inside a number field $k/\mathbb{Q}$:

Thm 3 Let $S$ be a Veech surface with a quadratic trace field (i.e., $r = [k: \mathbb{Q}] = 2$). Then the set of directions for which the directional flow is not even topologically weak mixing has positive Hausdorff dimension.

E.g. the regular polygon has trace field $k = \mathbb{Q}[\cos \frac{\pi}{n} ]$ Certainly then, the regular polygons have mysteries yet to unfold, but also I am curious what these positive Hausdorff dimension fractal sets might be. Another question is if there are Veech surfaces other than than translation surfaces of regular polygons.

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Some examples are given in the 2005 paper by Sergey Vasilyev.

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