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

Abstract

A Sturmian sequence is an infinite nonperiodic string over two letters with minimal subword complexity. In two papers, the first written by Morse and Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising correspondence was established between Sturmian sequences on one side and rotations by an irrational number on the unit circle on the other. In 1991 Arnoux and Rauzy observed that an induction process (invented by Rauzy in the late 1970s), related with the classical continued fraction algorithm, can be used to give a very elegant proof of this correspondence. This process, known as the Rauzy induction, extends naturally to interval exchange transformations (this is the setting in which it was first formalized). It has been conjectured since the early 1990s that these correspondences 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 counterexamples like the one constructed in 2000 by Cassaigne, Ferenczi, and Zamboni are rare. It is the aim of the present chapter to survey all these ideas and results.