On the Genesis of Symbolic Dynamics as We Know It

We trace the beginning of symbolic dynamics—the study of the shift dynamical system—as it arose from the use of coding to study recurrence and transitivity of geodesics. It is our assertion that neither Hadamard’s 1898 paper, nor the Morse-Hedlund papers of 1938 and 1940, which are normally cited as the first instances of symbolic dynamics, truly present the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the beginning of symbolic dynamics in a paper published by Hedlund in 1944. Symbolic dynamics, in the modern view [LM95, Kit98], is the dynamical study of the shift automorphism on the space of bi-infinite sequences of symbols, or its restriction to closed invariant subsets. In this note, we attempt to trace the beginnings of this viewpoint. While various schemes for symbolic coding of geometric and dynamic phenomena have been around at least since Hadamard (or Gauss: see [KU05]), and the two papers by Morse and Hedlund entitled “Symbolic dynamics” [MH38, MH40] are often cited as the beginnings of the subject, it is our view that the specific, abstract version of symbolic dynamics familiar to us today really began with a paper,“Sturmian minimal sets” [Hed44], published by Hedlund a few years later. The outlines of the story are familiar, and involve the study of geodesic flows on surfaces, specifically their recurrence and transitivity properties; this note takes as its focus a letter from Hedlund to Morse, written between their joint papers and Hedlund’s, in which his intention to turn the subject into a part of topology is explicit. This letter is reproduced on page 6. Our focus here is rather narrow, even when it comes to coding geodesic flows. A recent survey by Katok and Ugarcovici [KU05] distinguishes two approaches to such coding: a geometric method, which is our subject, and a second, going back to Gauss and associated with Artin, Koebe and Nielsen which can be regarded as more arithmetic in nature. We do not propose to consider the latter in detail. The survey [KU05] discusses technical details of both approaches as well as their subsequent development in recent years. The beginnings of symbolic dynamics are often traced back to Hadamard’s 1898 study of geodesics on surfaces of negative curvature [Had98]. In Part II of this work (§20), Hadamard gives a coding for the (free homotopy classes of) closed geodesics, essentially as words (up to cyclic permutation) in generators of the fundamental group, and in Part III (§37) he shows that each word corresponds to a unique 2000 Mathematics Subject Classification. Primary 37B10; Secondary 01A60.