 
Summary: RECURSIVELY RENEWABLE WORDS AND CODING OF
IRRATIONAL ROTATIONS
SHIGEKI AKIYAMA AND MASAYUKI SHIRASAKA
Abstract. A word generated by coding of irrational rotation with respect to
a general decomposition of the unit interval is shown to have an inverse limit
structure directed by substitutions. We also characterize primitive substitutive
rotation words, as those having quadratic parameters.
1. Definitions and the results
Let A = {0, 1, . . . , m  1} be a finite set of letters and A
be the monoid over A
generated by concatenation, having the identity element , the empty word. The
set of right infinite words over A is denoted by AN
.
A sturmian word z is an element of AN
characterized by the property that
pz(n) = n + 1, where pz(n) is the number of factors (i.e. subwords) of length n
appears in z. The function pz(n) is called the complexity of z. Since pz(1) = 2, we
have A = {0, 1}. The sturmian word is known to have the lowest complexity among
aperiodic words. The aperiodicity implies that exactly one of {00, 11} appears in
z. Let us assume that 11 is forbidden in z. Then the sturmian word z = z0z1 · · ·
