Isoperiodic Dynamics in Rank 1 Affine Invariant Orbifolds

Ygouf, Florent

doi:10.1093/imrn/rnab153

International Mathematics Research Notices

2021

DOI: 10.1093/imrn/rnab153

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Isoperiodic Dynamics in Rank 1 Affine Invariant Orbifolds

Florent Ygouf

Abstract: Let ${\mathcal {M}}$ be a rank 1 affine invariant orbifold in a stratum of the moduli space of flat surfaces. We show that the leaves of the ${\mathcal {M}}$-isoperiodic foliation are either all closed or all dense. In the 2nd case, we establish ergodicity of the foliation with respect to the affine measure on ${\mathcal {M}}$.

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Cited by 2 publications

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Correction to: Rel leaves of the Arnoux–Yoccoz surfaces

Hooper

Weiss

2021

Sel. Math. New Ser.

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Our arguments in §6 of the paper [3] contain an error. In this note we explain the error and how to fix it. We are grateful to Florent Ygouf for both pointing out the mistake, and for indicating the correct argument included below. All of the results stated in the introduction of the paper remain valid, as a consequence of the amended argument which will be given below. In the recent preprint [4], Ygouf proves related results about existence of dense rel leaves in other loci.

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Correction to: Rel leaves of the Arnoux–Yoccoz surfaces

Hooper

Weiss

2021

Sel. Math. New Ser.

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Translation surfaces and periods of meromorphic differentials

Chenakkod

Gianluca

Gupta

2022

Proceedings of London Math Soc

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Let S$S$ be an oriented surface of genus g$g$ and n$n$ punctures. The periods of any meromorphic differential on S$S$, with respect to a choice of complex structure, determine a representation χ:Γg,n→double-struckC$\chi :\Gamma _{g,n} \rightarrow \mathbb {C}$ where normalΓg,n$\Gamma _{g,n}$ is the first homology group of S$S$. We characterise the representations that thus arise, that is, lie in the image of the period map sans-serifPer:normalΩMg,n→sans-serifHom(normalΓg,n,C)$\textsf {Per}:\Omega \mathcal {M}_{g,n}\rightarrow \textsf {Hom}(\Gamma _{g,n}, {\mathbb {C}})$. This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S$S$ with the prescribed holonomy χ$\chi$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

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Isoperiodic Dynamics in Rank 1 Affine Invariant Orbifolds

Cited by 2 publications

References 8 publications

Correction to: Rel leaves of the Arnoux–Yoccoz surfaces

Correction to: Rel leaves of the Arnoux–Yoccoz surfaces

Translation surfaces and periods of meromorphic differentials

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