Generalized eigenfunctions of interval exchange maps

Abstract

L etf 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, � n(T ), of the partition defined by T n is of the order 1/n for infinitely many n. This assumption is generic. Given δ> 0 we prove that for sufficiently large n and for every interval J satisfying |J |= � n(T ), there exists x0 ∈ J such that |{x ∈ J :| f( x)− f( x0) |≥ δ}| <δ |J |. This provides a specific sufficient generic diophantine condition for Veech's result (V2, Lemma 7.3). Above |·| 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)= γ exp(2πiax)almost everywhere, where γ ∈ C and a ∈ R.