# Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms

@inproceedings{Winsor2021DynamicsOT, title={Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms}, author={Karl Winsor}, year={2021} }

Let Sg be a closed oriented surface of genus g, and let ΩMg(κ) be a stratum of the moduli space of holomorphic 1-forms of genus g. We show that the absolute period foliation of ΩMg(κ) is ergodic on the area-1 locus, and we give an explicit full measure set of dense leaves, subject to a mild constraint on κ. For an algebraically generic cohomology class φ ∈ H(Sg;C), we show that the space of holomorphic 1-forms in ΩMg(κ) representing φ is connected. On the other hand, in the non-generic case, we… Expand

#### One Citation

DENSE REAL REL FLOW ORBITS AND ABSOLUTE PERIOD LEAVES

- 2021

We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, we… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

Moduli spaces of isoperiodic forms on Riemann surfaces

- Mathematics
- 2014

This paper describes the intrinsic geometry of a leaf A(L) of the absolute period foliation of the Hodge bundle ΩMg: its singular Euclidean structure, its natural foliations and its discretized… Expand

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

- Mathematics
- 2002

Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe… Expand

Foliations of Hilbert modular surfaces

- Mathematics
- 2007

The Hilbert modular surface XD is the moduli space of Abelian varieties A with real multiplication by a quadratic order of discriminant D > 1. The locus where A is a product of elliptic curves… Expand

A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds

- Mathematics
- 2020

We define on any affine invariant orbifold M a foliation F^M that generalises the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in… Expand

The field of definition of affine invariant submanifolds of the moduli space of abelian differentials

- Mathematics
- 2014

The field of definition of an affine invariant submanifold M is the smallest subfield of R such that M can be defined in local period coordinates by linear equations with coefficients in this field.… Expand

DENSE REAL REL FLOW ORBITS AND ABSOLUTE PERIOD LEAVES

- 2021

We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, we… Expand

Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

- Mathematics
- 2002

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment… Expand

Periods of Abelian Differentials With Prescribed Singularities

- Mathematics
- 2020

We give a necessary and sufficient condition for a representation of the fundamental group of a closed surface of genus at least 2 to $\mathbb C$ to be the holonomy of a translation surface with a… Expand

Isoperiodic dynamics in rank 1 affine invariant orbifolds

- Mathematics
- 2021

Let 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 M-isoperiodic foliation are either all closed or all dense. In the second… Expand

Dynamics of SL 2 ( R ) over moduli space in genus two

- 2003

This paper classifies orbit closures and invariant measures for the natural action of SL2(R) on ΩM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.