Tight upper bounds on the number of invariant components on translation surfaces

Abstract

An abelian differential on a surface defines a flat metric and a vector field on the complement of a finite set of points. The vertical flow that can be defined on the surface has two kinds of invariant closed sets (i.e. invariant components) -periodic components and minimal components. We give upper bounds on the number of minimal components, on the number of periodic components and on the total number of invariant components in every stratum of abelian differentials. We also show that these bounds are tight in every stratum.

Différentielles abéliennes à singularités prescrites

“…-La seconde application est sur la géométrie des surfaces de translations associées à une différentielle abélienne holomorphe. Naveh a montré dans [Nav08] que le nombre maximal de cylindres disjoints dans une différentielle holomorphe de la strate ΩM g (a 1 , . .…”

Geometry of periodic regions on flat surfaces and associated Siegel–Veech constants

“…To prove this result we need to determine the maximal number of cylinders that can come from any configuration which is admissible for a fixed stratum H (α). We insist on the fact that we compute here the number of cylinders in rigid collections of saddle connections, which is different from the studies of Naveh [15] and Lindsey [16] where they count the number of parallel cylinders.…”

“…Their distinguishing feature is that they persist under any small deformation. For related counting problems in the case of general periodic components (so not configurations), see [15,16].…”

Counting saddle connections in flat surfaces with poles of higher order

“…Theorem 2 of [25] proves that for classical translation surfaces, the maximal number of components invariant by the vertical flow is g + n − 1. This bound is sharp.…”