A criterion for weak mixing of induced interval exchange transformations

TL;DR

This paper proves that for a typical interval exchange transformation, the set of subintervals inducing weakly mixing transformations is large, with full measure, based on a generic Diophantine condition on the subinterval length.

Contribution

It establishes a generic Diophantine criterion ensuring weak mixing for induced interval exchange transformations, showing this property holds for a residual set of subintervals.

Findings

The set of t for which the induced IET is weakly mixing has full Lebesgue measure.

A Diophantine condition on t guarantees weak mixing of the induced IET.

Weak mixing is generic among induced IETs for ergodic IETs.

Abstract

Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show that \[ \{0<t<1\mid f_{t} \text{is weakly mixing}\} \] is a residual subset of $X$ of full Lebesgue measure. The result is proved by establishing a generic Diophantine sufficient condition on $t$ for $f_{t}$ to be weakly mixing.