Leaves of laminations can be symbolically represented by deforming them into paths of labeled embedded carrier graphs, including train tracks. Here, we describe and characterize the languages of two-way infinite words coming from this kind of coding, called lamination languages, first, by using carrier graph sequences, and second, by using word combinatorics. These characterizations generalize those existing for interval exchange transformations. We also show that lamination languages have ultimately affine factor complexity, and we present effective techniques to build these languages.

No abstract

No abstract

Abstract. We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties. IntroductionInterval exchange transformations (IETs) are maps over I = [0, 1) which can be defined as permutations of intervals partitioning I . In the finite case, these maps are just piecewise isometries of I onto itself, or equivalently, injective measure-preserving maps having only a finite number of discontinuities. They happen to be a fundamental notion in dynamical systems and ergodic theory [CFS82,Mañ87,HK02]. Another important notion in the same context is the shift σ on the set A N of infinite words (symbolic sequences) over a finite alphabet A. This simple continuous map consists of erasing the first letter of its argument. The pair (A N , σ ) forms a basic topological dynamical system, where A N is then called the full shift over A. If L is a closed σ -invariant subset of A N , the pair (L , σ ) also induces such a system, where L is then just called a shift [LM95, Kit98]. The topological entropy of a shift L depends on the factor complexity (block growth) of L [MH38, Par66, CN10], i.e. the map p L on N * giving for each n the number of distinct length-n factors (subblocks) occurring in the words of L. Now, a known general relationship between all the above concepts is the following: the support I of a finite IET can be embedded as a subset into a measured compact space in such a way that the IET extends to a measurepreserving continuous map, whose natural symbolic conjugate is a shift L with a factor complexity bounded by an affine function (thus L has zero topological entropy) [Kea75]. The main idea of this paper is to study this relationship the other way around, that is,

No abstract