The Periodic Points of Morse-Smale Endomorphisms of the Circle

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