GL2ℝ–invariant measures in marked strata : generic marked points, Earle–Kra for strata and illumination

Abstract

We classify GL(2, R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmüller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2, R) orbit.

Existence of closed geodesics through a regular point on translation surfaces

“…Marked points and periodic points. A point p on a trans- 4,6,7,14,28]). For a Veech surface, this is equivalent to saying that the orbit of p under Aff(X, ω) is finite.…”

“…(1) The strategy to prove Theorem 1 is to show that any regular point in a translation surface (X, ω) that is not contained in any simple closed geodesic is a "periodic point" in the sense of Apisa [4].…”

“…If L = H hyp (κ), then by [4,Theorem 1.5], p must be a regular Weierstrass point of X. It follows from Proposition 6.1 that p is contained in a simple cylinder of (X, ω).…”

Periodic points on the regular and double $n$-gon surfaces

“…Remark 1.2. Our definition is equivalent to the one used in Apisa [Api20]. A version of this definition which includes the zeros of ω first appeared in Gutkin-Hubert-Schmidt [GHS03].…”

“…First, we use the transfer principle to reduce the problem to classifying periodic points on an explicit set of saddle connections. Second, we classify the periodic points on these saddle connections by covering them with two collections of non-parallel cylinders and using the "rational height lemma" of Apisa [Api20], which we will recall.…”