Periodicity, morphisms, and matrices

In 1965, Fine and Wilf proved the following theorem: if (fn)n¿0 and (gn)n¿0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn = gn for 06 n¡h+ k − gcd(h; k), then fn = gn for all n¿ 0. Furthermore, the constant h + k − gcd(h; k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn 6 gn instead of fn=gn. We also obtain generalizations to more than two periods. We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if h : ∗ → ∗ is a morphism with | | = n, and w is a word with |w|¿ |h(w)|¿ |h2(w)|¿ · · ·¿ |hk(w)|, then k6 n. c © 2002 Elsevier Science B.V. All rights reserved.