Nondegeneracy of the spectrum of the twisted cocycle for interval exchange transformations

TL;DR

This paper proves the positivity of the top Lyapunov exponent for the twisted cocycle associated with interval exchange transformations, linking it to affine invariant submanifolds and providing applications in spectral theory.

Contribution

It establishes the nondegeneracy of the spectrum for the twisted cocycle of IETs and connects the top exponent to limits along affine invariant submanifolds, advancing understanding in spectral dynamics.

Findings

Positivity of the top Lyapunov exponent for the twisted cocycle.

Relation of the exponent to limits along affine invariant submanifolds.

Applications to a conjecture of Kontsevich and Zorich, discrepancy estimates, and spectral measure dimensions.

Abstract

We prove the positivity of the top Lyapunov exponent of the twisted (spectral) cocycle, associated with IETs, with respect to a family of natural invariant measures. The proof relies on relating the top exponent to limits of exponents along families of affine invariant submanifolds of genus tending to infinity. Applications include an observation about a conjecture of Kontsevich and Zorich, a discrepancy estimate, and a formula for the lower local dimension of spectral measures.