Periodic orbits of large diameter for circle maps

Let f be a continuous circle map and let F be a lifting of f. In this paper we study how the existence of a large orbit for F affects its set of periods. More precisely, we show that, if F is of degree d ≥ 1 and has a periodic orbit of diameter larger than 1, then F has periodic points of period n for all integers n ≥ 1, and thus so has f. We also give examples showing that this result does not hold when the degree is nonpositive. © 2010 American Mathematical Society.