On the ergodicity of anti-symmetric skew products with singularities and its applications

TL;DR

This paper presents a new method for proving ergodicity of skew products involving interval exchange transformations with singularities, extending applicability beyond logarithmic cases and aiding spectral analysis of flows.

Contribution

Introduces a novel approach inspired by Borel-Cantelli arguments that handles broader singularities and improves ergodicity proofs for IET skew products.

Findings

Method applies to singularities beyond logarithmic type.

Effective for symmetric IETs and antisymmetric cocycles.

Facilitates study of spectral error terms in Hamiltonian flows.

Abstract

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lema\'nczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.