Veech Groups, Cutting Sequences, and Penetration Spectra - C. Ulcigrai

Introduzione
Translation surfaces Translation surfaces are obtained by glueing polygons with pairs of parallel sides identified by translations
Affine automorphisms and Veech groups Definition (Affine diffeomorphisms) A diffeomorphism S S between translation surfaces S. S' is affine iff it is affine in each chart and Du(z) is independent onze S. Definition (Veech group)
Veech (or lattice) surfaces
Cutting sequences of linear trajectories Givena a polygonal presentation of a (Veech) translation surface, label by an alphabet A pairs of glued sides
Brief history of cutting sequences characterizations
A characterization of Sturmian sequences via substitutions
Properties of the classical Lagrange spectrum
Hallray for Veech translation surfaces We generalize the result by Hall to any Veech translation surface
Affine deformations and Teichmueller curves Let S be a Veech surface (V(56) is a lattice)
Ideas of proofs of cutting sequences characterizations
The octagon: other transitions diagrams The action of gives labels permutations and transition diagrams D, for
Derived sequences Definition The derived sequence of an admissible quence is obtained by keeping followed by the same letter Example
Geometric interpreation of derivation for the octagon
Renormalization on the octagon Teichmüller disk
Bouw-Moeller derivation Given a cutting sequence w for Sa... we define a combinatorial derivation
Renormalization on the Bouw Moeller Teichmüller disk
Hall Rays in Lagrange Spectra of Veech surfaces Let S be a Veech surface. Consider a fundamental domain for V(5)
Hall Rays in Lagrange Spectra of Veech surfaces Let S be a Veech surface. Consider a fundamental domain for V(S).
Boundary expansions and wedges of saddle connections