Symbolic Dynamics II. Sturmian Trajectories

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.1. Introduction. In a recent paper 1 we initiated a theory of symbolic dynamics. In this theory we consider unending sequences of symbols or symbolic trajectories and devote attention to those properties of symbolic trajectories which are suggested by dynamical considerations. A symbolic trajectory is formed from symbols taken from a finite set of generating symbols subject to certain rules of admissibility. In SD admissibility conditions were formulated of such generality that the resulting symbolic trajectories include in particular those which arise in the geodesic problem on surfaces which satisfy the condition of uniform geodesic instability.However, no surface of the topological type of a torus satisfies the conclition of uniform geodesic instability and the admissibility conditions of SD do not include those which arise in the case of the torus.In the present paper we consider a class of symbolic trajectories formed from two generating symbols subject to admissibility conditions defined by a simple comparison property. These are the symbolic trajectories which characterize the geodesics on a flat torus. They may be used to characterize the distribution of the zeros of the solutions of a differential equation of the form y" + f(x)y = 0, where f(x) is a periodic function of x. We term the trajectories of this class Sturmian. A first fundamental result is as follows:Sturmian trajectories possess certain numerical characteristics, namely, a frequency, a pole, and a type index, and admit mechanical constructions uniquely determined by these characteristics.There are three types of Sturmian trajectories,-irrational, skew and periodic. The trajectories of irrational type are recurrent but not periodic; those of skew type are not recurrent. The recurrency functioll of a recurrent Sturmian trajectory is completely determined by the frequency a of the trajectory and may be denoted by R(n, a). We introduce the variable y = 2 (1 + a)-'. Let CG/Dv be the convergents in a continued fraction representation of y. We have the following fundamental theorem:

Initial powers of Sturmian sequences

“…Sturmian sequences are exactly those one-sided infinite sequences with complexity p(n) = n + 1 for every n (see [34,14]). Write X α for the set of all Sturmian sequences of slope α, and denote by T the shift map on sequences, i.e., (T (ω)) i = ω i+1 .…”

“…since [34], though Hedlund and Morse did not address this question specifically. One can also obtain the formula for ice of a characteristic sequence using Cassaigne's formula for the recurrence quotient in [13].…”

On Integer Sequences Generated by Linear Maps

“…. Clearly, the function P(w, n) is non-decreasing in n. It is bounded from above by an absolute constant independent of n if and only if the sequence w is ultimately periodic; otherwise, P(w, n) n + 1 for each positive integer n (see [14] or [15]). The sequences w for which equality P(w, n) = n + 1 holds for each positive integer n are called Sturmian sequences (see [3,4,14]).…”

Optimization and majorization of invariant measures

“…Indeed the family S is well known, having been studied, in various contexts and for varying reasons, since the early days of symbolic dynamics: Sturmian measures, or rather the symbol sequences (Sturmian sequences 2 ) corresponding to points in their support, were defined by Morse & Hedlund [40]. For each ∈ (0, 1) there is a unique non-empty minimal closed invariant set X on which T is combinatorially equivalent to rotation by angle ; the set X is a periodic orbit if and only if is rational (see e.g.…”