Ergodicity of explicit logarithmic cocycles over IETs

TL;DR

This paper proves ergodicity for a class of skew-product extensions over interval exchange transformations with logarithmic singularities, providing explicit examples and general conditions for ergodicity.

Contribution

It establishes ergodicity for a broad class of cocycles with logarithmic singularities over IETs, including explicit examples and general conditions based on permutation and length vectors.

Findings

Explicit ergodic examples of $ ext{R}$-extensions of Hamiltonian flows.

Ergodicity holds for almost every length vector under certain symmetry conditions.

Provides a criterion for ergodicity based on cocycle symmetry and permutation properties.

Abstract

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic $\mathbb{R}$-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths vector, the skew-product built over the IET with the given permutation and lengths vector given by a cocycle, with symmetric, logarithmic singularities, which is \emph{odd} when restricted to each continuity subinterval is ergodic.