958 research outputs found
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
The Lagrange spectrum of a Veech surface has a Hall ray
We study Lagrange spectra of Veech translation surfaces, which are a
generalization of the classical Lagrange spectrum. We show that any such
Lagrange spectrum contains a Hall ray. As a main tool, we use the boundary
expansion developed by Bowen and Series to code geodesics in the corresponding
Teichm\"uller disk and prove a formula which allows to express large values in
the Lagrange spectrum as sums of Cantor sets.Comment: 30 pages, 5 figures. Minor revisio
Convex lattice polygonal lines with a constrained number of vertices
A detailed combinatorial analysis of planar convex lattice polygonal lines is
presented. This makes it possible to answer an open question of Vershik
regarding the existence of a limit shape when the number of vertices is
constrained
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