Sébastien Ferenczi

Complexity of sequences and dynamical systems

Abstract This is a survey of recent results on the notion of symbolic complexity, which counts the number of factors of an infinite sequence, particularly in view of its relations with dynamical systems.

Structure of three-interval exchange transformations II: a combinatorial description of the tranjectories

AstractWe describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence, we obtain a complete characterization of those sequences of block complexity 2n+1 which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy.

Measure-theoretic complexity of ergodic systems

AbstractWe define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small

Complexity for finite factors of infinite sequences

\bar d

Tiling and local rank properties of the Morse sequence

-balls aroundα-n-names necessary to cover most of the system, for any generating partitionα. We show that this rate is essentially bounded if and only if the system is a translation of a compact group, and compute it for several classes of systems of entropy zero, thus getting examples of growth rates inO(n),O(nk) fork ε ℕ, oro(f(n)) for any given unboundedf, and of various relationships with the usual notion of language complexity of the underlying topological system.

A self-dual induction for three-interval exchange transformations

In this paper, we present a detailed study of the spectral/ergodic properties of three-interval exchange transformations. Our approach is mostly combinatorial and relies on the diophantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure-theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previously known examples, allow us to exhibit new examples of weakly mixing three-interval exchanges. Finally, we give affirmative answers to two questions posed by W. A. Veech on the existence of three-interval exchanges having irrational eigenvalues and discrete spectrum.

Combinatorial trees arising in the study of interval exchange transformations

Abstract We define several notions of language complexity for finite words, and use them to define and compute some new complexity functions for infinite sequences. In particular, they give a new characterization of Sturmian sequences and discriminate between Sturmian sequences with bounded or unbounded partial quotients.

A density theorem for (multiplicity, rank) pairs

We study a particular case of the two-dimensional Steinhaus theorem, giving estimates of the possible distances between points of the formkα andkα+β on the unit circle, through an approximation algorithm of β by the pointskα. This allows us to compute covering numbers (maximal measures of Rokhlin stacks having certain prescribed regularity properties) for the symbolic dynamical systems associated to the rotation of argument α, acting on the partition of the circle by the points 0, β. We can the compute topological and measure-theoretic covering numbers for exchange of three intervals; in this way, we prove that every ergodic exchange of three intervals has simple spectrum and build a new class of three-interval exchanges which are not of rank one.