A transfer principle: from periods to isoperiodic foliations

Holonomy of complex projective structures on surfaces with prescribed branch data

“…Indeed, if g ⩾ 2 and 𝜒 ∈ Hom(Γ, ℂ) is the holonomy of a translation surface such that Λ = 𝜒(Γ) is a lattice, then 𝑑 = Vol(𝜒)∕Area(Λ) must be at least 2 because the developing map induces a branched cover Σ → ℂ∕Λ of degree 𝑑. This obstruction was generalized by [4] for translation surfaces with prescribed singularities, where the authors observed that 𝑑 must satisfy 𝑑 ⩾ max 𝑖 𝑛 𝑖 + 1 and asked if these were the only obstructions to being the holonomy of a translation surface with prescribed singularities. This latter question was answered in the positive in [1,27].…”

“…As [𝜌(𝑎 2 ), 𝜌(𝑏 2 )] = id, the group generated by 𝜌(𝑎 2 ) and 𝜌(𝑏 2 ) is either cyclic or K. It cannot be K because one can check that 𝔖 4 is not generated by K together with a single 𝜎 ∈ 𝔖 4 . As the group generated by 𝜌(𝑎 2 ) and 𝜌(𝑏 2 ) is cyclic, we may assume that 𝜌(𝑏 2 ) = id and, interchanging the handle, that 𝜌(𝑎 1 ) ∈ 𝔄 4 .…”

“…Indeed, if $$g\geqslant 2$$ and $$\chi \in \mathrm{Hom}(\Gamma , \mathbb {C})$$ is the holonomy of a translation surface such that $$\Lambda = \chi (\Gamma )$$ is a lattice, then $$d=\mathrm{Vol}(\chi ) /\mathrm{Area}(\Lambda )$$ must be at least 2 because the developing map induces a branched cover $$\Sigma \rightarrow \mathbb {C}/\Lambda$$ of degree $$d$$. This obstruction was generalized by [4] for translation surfaces with prescribed singularities, where the authors observed that $$d$$ must satisfy $$d\geqslant \max _i n_i +1$$ and asked if these were the only obstructions to being the holonomy of a translation surface with prescribed singularities. This latter question was answered in the positive in [1, 27].…”

On the ergodic theory of the real Rel foliation

“…The first ergodicity results for the full Rel foliation were obtained by McMullen [McM], who proved ergodicity in the two strata and . Subsequently, Calsamiglia, Deroin and Francaviglia [CDF] proved ergodicity in all principal strata, and Hamenstädt [Ham] reproved their result by a simpler argument. Recently, Winsor [Wi1] proved ergodicity for most of the additional strata and, in [Wi2], showed that there are dense orbits for the -flow, for any as in Theorem 1.1.…”

A criterion for density of the isoperiodic leaves in rank one affine invariant orbifolds

“…Shortly after, Calsamiglia, Deroin, and Francaviglia have obtained a Ratner‐like classification of the minimal sets in the principal stratum and obtained the ergodicity with respect to the Masur–Veech measure as a consequence of this classification. See [7]. Simultaneously, Hamenstädt gave another proof of the ergodicity in the principal strata in [13].…”