The periodic points of Morse-Smale endomorphisms of the circle

Abstract:

Let $MS({S^1})$ denote the set of continuously differentiable maps of the circle with finite nonwandering set, which satisfy certain generic properties. For $f \in MS({S^1})$ let $P(f)$ denote the set of positive integers which occur as the period of some periodic point of f. It is shown that for $f \in MS({S^1})$ there are integers $m \geqslant 1$ and $n \geqslant 0$ such that $P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}$. Conversely, if m and n are integers, $m \geqslant 1,n \geqslant 0$, there is a map $f \in MS({S^1})$ with $P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}$.