Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux-Yoccoz map

Abstract

In this article we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals and semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux-Yoccoz interval exchange map satisfies these conditions.

34 pages, submitted to Ergodic Theory and Dynamical Systems

References in corpus (3)

• Affine interval exchange maps with a wandering interval
• Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus one
• Geometric representation of interval exchange maps over algebraic number fields