Ergodicity of the absolute period foliation
One can define translation surfaces as surfaces obtained by gluing parallel sides of a polygon: namely if \(e_1, \ldots, e_n\) are nonzero vectors in \(\mathbb C\) and \(P\) is a polygon with sides \(e_1, \ldots, e_n\) each occuring exactly twice (in no particular order) then identifying the two sides supported by \(e_i\) for all \(i\) gives a closed surface \(S\). It has the additional structure of a flat metric, possibly with conical points with angles in \(2\pi\mathbb N\), at the vertices of \(P\); to simplify let us assume that all vertices give rise to cone points. Moreover singular chains coming from \(e_1, \ldots, e_n\) give a basis for the relative homology \(H_1(S, V)\) where \(V\) is the set of vertices of \(P\) giving rise to cone points. There is a natural decomposition of the moduli space of such surfaces into strata corresponding to the orders of cone points. Generically there are only cone points of angle \(4\pi\) (the minimal possible); the stratum corresponding to such surfaces is called the principal stratum. It is also possible to define a translation surface as a pair \((X, \omega)\) where \(X\) is a Riemann surface and \(\omega\) a holomorphic differential (also called an ``abelian differential''). The angle of the cone points then corresponds to the order of zeroes of \(\omega\). Integrating \(\omega\) against the relative cycles defined above gives local coordinates for the moduli space of such surfaces. The absolute homology \(H_1(S)\) injects into \(H_1(S, V)\). Integrals against absolute classes are called absolute periods, and the absolute period foliation is given by level sets of absolute periods; that is a leaf is a maximal subset where all such periods are constant. Such leaves are contained in strata of the decomposition described above. \textit{C. T. McMullen} [Duke Math. J. 163, No. 12, 2271--2323 (2014; Zbl 1371.30037)] proved that if \(S\) has genus 2 or 3 this foliation is ergodic on the principal stratum, meaning that every Borel subset which is a union of leaves either has null measure with respect to the Lebesgue measure class, or its complement does. In the present paper the author generalises this result to all \(S\) of genus \(\geq 2\).
scientific article; zbMATH DE number 6898289