Weak Mixing Property for Linear Involutions

Abstract

In this work, we extend the celebrated result of Avila--Forni~\cite{avila2007weak} on the weak mixing property of interval exchange transformations to the setting of linear involutions, which naturally arise from the study of vertical foliations on half-translation surfaces. Using recent advances on the Kontsevich--Zorich cocycle for quadratic differentials~\cite{belldiagonal, gutierrez2019classification, trevino2013non}, we establish that, for every dynamically irreducible generalized permutation, the associated linear involution is weakly mixing for almost every admissible parameter.