Generalized eigenfunctions of interval exchange maps

Abstract

Let f be a generalized eigenfunction of an interval exchange transformation, T, on the unit interval, which satisfies the infinite distinct orbit condition (IDOC). We assume that the minimum spacing, $\epsilon_n(T)$, of the partition defined by Tⁿ is of the order 1/n for infinitely many n. This assumption is generic. Given $\delta>0$ we prove that for sufficiently large n and for every interval J satisfying $\vert J \vert =\epsilon_n(T)$, there exists $x_0 \in J$ such that \[\vert \{ x \in J : \vert f(x) - f(x_0) \vert \geq \delta \} \vert< \delta \vert J \vert.\] This provides a specific sufficient generic diophantine condition for Veech's result [V2, Lemma 7.3]. Above $\vert \cdot\vert$ denotes the linear measure of the set. Moreover, if T is uniquely ergodic, then any bounded variation continuous f must be of the form $f(x)= \gamma \exp(2 \pi iax)$ almost everywhere, where $\gamma \in \mathbb{C}$ and $a \in \mathbb{R}$.