The set of maps F_{a,b}: x -> x+a+{b/{2 pi}} sin(2 pi x) with any given rotation interval is contractible

Abstract

Consider the two-parameter family of real analytic maps $F_{a,b}:x\mapsto x+a+\frac{b}{2\pi }\sin (2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set.