\(\boldsymbol{S}\)-adic sequences. A bridge between dynamics, arithmetic, and geometry

Abstract

A Sturmian sequence is an infinite string over two letters with low subword complexity. The history of the research surveyed in the present chapter starts with two papers written by Morse and Hedlund as well as Coven and Hedlund in 1940 and 1973, respectively. In these papers the authors established a surprising correspondence between Sturmian sequences and rotations by an irrational number on the unit circle. Later, Arnoux and Rauzy observed that an induction process in which the classical continued fraction algorithm appears can be used to give another very elegant proof of this correspondence. It has been conjectured since the early 1990ies that these correspondences between rotations on the unit circle, continued fractions, and Sturmian sequences carry over to rotations on higher dimensional tori, generalized continued fraction algorithms, and so-called \(S\)-adic sequences generated by substitutions. The idea of working towards such a generalization is known as Rauzy's program. Recently Berth\'e, Steiner, and Thuswaldner made some progress on Rauzy's program and were indeed able to set up the conjectured generalization of the above correspondences. Using a generalization of Rauzy's induction process in which generalized continued fraction algorithms show up, they proved that under certain natural conditions an \(S\)-adic sequence gives rise to a dynamical system which is measurably conjugate to a rotation on a higher dimensional torus. Moreover, they established a metric theory which shows that exceptional cases like the ones constructed in 2000 and 2006 by Cassaigne and his coauthors are rare. It is the aim of the present paper to survey all these ideas and results.