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

Abstract We introduce a novel method for proving the ergodicity of 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 given by Fayad and Lemańczyk On the ergodicity of cylindrical transformations given by the logarithm. Mosc. Math. J. 6 (2006), 657–672. 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 anti-symmetric cocycles. Moreover, its most significant advantage is the 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.