Related papers: Most Interval Exchanges Have No Roots
We study the group of interval exchange transformations. Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged subintervals. We…
In this paper, we prove a criterion for existence of continuous non constant eigenfunctions for interval exchange transformations, that is for non topologically weak mixing. We first construct, for any m>3, uniquely ergodic interval…
A disjoint rotation map is an interval exchange transformation (IET) on the unit interval that acts by rotation on a finite number of invariant subintervals. It is currently unknown whether the group E of all IETs possesses any non-abelian…
We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval…
This paper uses a construction of M. Keane to show that there exists a topologically mixing interval exchange transformation.
Interval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange…
Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In…
An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of…
We study the group of all interval exchange transformations (IETs). We show that for every IET $S$, there exists a dense open set of admissible IETs that share a relation with $S$. This is an extension of a result published by Dahmani,…
For irreducible interval exchange transformations, we study the relation between the powers of induced map and the induced maps of powers and raise a condition of equivalence between them. And skew production of Rauzy induction map is set…
An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations…
We study the group IET of all interval exchange transformations. Our first main result is that the group generated by a generic pairs of elements of IET is not free (assuming a suitable irreducibility condition on the underlying…
We prove that any over-twist pattern is conjugate to an interval exchange transformation with bounded number of segments of isometry, restricted on one of its cycles. The bound is independent of the period and over-rotation number of the…
In this paper we study the non-injectivity arising in infinite interval exchange transformations. In particular, we build and analyze an infinite family of infinite interval exchanges semi-conjugated to generalized Thue-Morse subshifts,…
A sharp bound on the number of invariant components of an interval exchange transformation is provided. More precisely, it is proved that the number of periodic components n_per and the number of minimal components n_min of an interval…
Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show…
In [Mas82] and [Vee78] it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the…
In this paper we investigate translated cone exchange transformations, a new family of piecewise isometries and renormalize its first return map to a subset of its partition. As a consequence we show that the existence of an embedding of an…
In this article we prove that given a self-similar interval exchange transformation T, whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals…
We study an interval exchange transformation of [0,1] formed by cutting the interval at the points 1/n and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero.…