2019/08/16 by Jörg Μ. Thuswaldner, Thuswaldner, Jörg M.
Computer Science · Mathematics · #Cellular Automata and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1908.05954
openalex publication_date 2019/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Sturmian sequence is an infinite nonperiodic string over two letters with\nminimal subword complexity. In two papers, the first written by Morse and\nHedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising\ncorrespondence was established between Sturmian sequences on one side and\nrotations by an irrational number on the unit circle on the other. In 1991\nArnoux and Rauzy observed that an induction process (invented by Rauzy in the\nlate 1970s), related with the classical continued fraction algorithm, can be\nused to give a very elegant proof of this correspondence. This process, known\nas the Rauzy induction, extends naturally to interval exchange transformations\n(this is the setting in which it was first formalized). It has been conjectured\nsince the early 1990s that these correspondences carry over to rotations on\nhigher dimensional tori, generalized continued fraction algorithms, and\nso-called S-adic sequences generated by substitutions. The idea of working\ntowards such a generalization is known as Rauzy's program. Recently Berth 'e,\nSteiner, and Thuswaldner made some progress on Rauzy's program and were indeed\nable to set up the conjectured generalization of the above correspondences.\nUsing a generalization of Rauzy's induction process in which generalized\ncontinued fraction algorithms show up, they proved that under certain natural\nconditions an S-adic sequence gives rise to a dynamical system which is\nmeasurably conjugate to a rotation on a higher dimensional torus. Moreover,\nthey established a metric theory which shows that counterexamples like the one\nconstructed in 2000 by Cassaigne, Ferenczi, and Zamboni are rare. It is the aim\nof the present chapter to survey all these ideas and results.\n