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.

Characterization of minimal sequences associated with self-similar interval exchange maps

“…We adopt all notations of Section 2.1 and we assume β is a non-real eigenvalue of M = M σ with |β| > 1. In Lemma 4.4 in [CGM17] it is proved that:…”

“…We also consider F Notice that v a,(p,c,s) (τ ) ≥ v a (τ ) and v a (τ ) = min (p,c,s)∈Āa v a,(p,c,s) (τ ). In Lemmas 5.3 and 7.2 in [CGM17] it is proved that:…”

“…This condition is also equivalent to the fact that log is generated by eigenvectors different from the Perron-Frobenius eigenvector of the matrix M associated with T . In subsequent works [CG97], [Cob02], [BHM10], [BBH14] and [CGM17], the existence of such an affine i.e.m. having wandering intervals is shown to be related to the spectral properties of M .…”

“…with slope vector = (exp(−γ a ); a ∈ A) which is semi-conjugate to T and has wandering intervals. Then, in [CGM17] this result was extended to the case of a non-real eigenvalue β with |β| > 1 such that β/|β| is not a root of unity and under the so called unique representation property for β and some associated eigenvector Γ (see Definition 2.5). More precisely, it was proved that for almost all complex eigenvectors γ in the complex vector space generated by Γ there exists an affine i.e.m.…”

“…Besides these two works, very little is known about them and understanding minimal sequences can shed light on such fractal sets. In this article we characterize the set of minimal sequences and provide a method to compute them assuming the same hypotheses as in [CGM17]. That is, β is a non-real eigenvalue of the matrix associated with the self-similar i.e.m.…”