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.

Abstract. Let X : R 2 → R 2 be a C 1 map. Denote by Spec(X) the set of (complex) eigenvalues of DX p when p varies in R 2 . If there exists ǫ > 0 such that Spec(X) ∩ (−ǫ, ǫ) = ∅, then X is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

Abstract. Given a compact 2-dimensional manifold M we classify all continuous flows ϕ without wandering points on M . This classification is performed by finding finitely many pairwise disjoint open ϕ−invariant subsets

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.

Abstract. It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.