Piece-wise affine maps conjugate to interval exchanges

Abstract

If T is an interval exchange transformation we denote by C_{{\rm aff}}(T) (respectively S_{{\rm aff}}(T)) the set of piece-wise affine maps of the interval which are conjugate (respectively semi-conjugate) to T. In this work we will give a description of the set C_{{\rm aff}}(T) for almost allT. We present an explicit interval exchange T_0 such that S_{{\rm aff}}(T_0)\backslash C_{{\rm aff}}(T_0) is non-empty. All the elements of S_{{\rm aff}}(T_0)\backslash C_{{\rm aff}}(T_0) are uniquely ergodic and have a unique wandering interval.

Affine interval exchange maps with a wandering interval

“…The infinite path in the Rauzy diagram in their case is periodic. The same example was studied more deeply by Cobo [5]. In particular, he put in evidence on this example the importance of the Oseledets decomposition of the extended Zorich cocycle (see Section 3.1 below).…”

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

“…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 .…”

One-dimensional contracting singular horseshoe

“…They involved some smoothness and other ingredients on the considered maps. In the present case we only require C 1 -smoothness perturbations, so wandering intervals may appear (see [22], [10], [15], [3], [12], [19]: Theorem 2.3, p. 43, [6], [11], [27] and [5]). Afterward, the structural stability will follow from the arguments in [14] (see section 4, pp.…”