STURMIAN EXPANSIONS AND ENTROPY E. ARTHUR ROBINSON, JR. 1. Introduction Sturmian sequences were introduced by Morse and Hedlund [10] as the sequences that code the orbits of the geodesic flow on a flat 2-torus. In this paper, we restrict our attention to (1-sided) aperiodic Sturmian sequences, which may be defined to be those sequences d = .d 1 d 2 d 3 ···∈{0, 1} N that have exactly n + 1 distinct factors (subsequences u = d j d j+1 ...d j+n−1 ) of length n. This property is often expressed by saying that a Sturmian sequence d has complexity function c d (n)= n + 1. If c e (n) is he complexity function of a sequence e ∈{0, 1} N , it is known (see [5], Chapter 6) that c e (k)= k for some k if and only if e is eventually periodic. Thus Sturmian sequences are the least complex among aperiodic sequences. A sequence d = .d 1 d 2 d 3 ···∈{0, 1} N is said to be balanced if for any i, j, ℓ ≥ 1       i+ℓ−1  k=i d k − j+ℓ−1  k=j d k       ≤ 1. It can be shown that a sequence is balanced if and only if it is Sturmian, and from this, one can prove (see [5], Chapter 6) that the limit (1) α = lim n→∞ 1 n ∞  k=1 d k exists, and α is irrational. The number α is called the slope of d. Morse and Hedlund [10] showed that if d = .d 1 d 2 d 3 ... is a Sturmian sequence with slope α ∈ (0, 1)\Q, then there is a unique x ∈ [0, 1), called the intercept, so that d either has the form (2) d n = ⌊α(n + 1) + x⌋−⌊αn + x⌋, for all n ∈ N, or (3) d n = ⌈α(n + 1) + x⌉−⌈αn + x⌉. Note that (2) and (3) are the same unless nα + x = 0 mod 1 for some n> 1, in which case they disagree in exactly one or two adjacent digits. Given a Sturmian sequence d, one can easily determine its slope α using (1). The goal of this paper is to exhibit a similarly simple formula for the intercept x. In particular, the intercept x can be obtained using a well know generalization of continued fraction and radix expansions, called an f -expansions. Another way to say this is that a Sturmian sequence d = .d 1 d 2 d 2 ··· ∈ {0, 1} N can be regarded Date : October 15, 2010. 2010 Mathematics Subject Classification. 11K16, 28D20, 37E05, 37B10. Key words and phrases. numeration, entropy, interval maps, symbolic dynamics. 1