Roots of Continuous Piecewise Monotone Maps of an Interval

We shall consider slightly more general problems. Namely, we shall investigate the existence of continuous: piecewise monotone, piecewise strictly monotone, and piecewise linear n-th roots of interval maps which have a continuous n-th root. Here by an n-th root of f we mean a map g such that f = g (g is the n-th iterate of g). A continuous map f : I → J , where I, J are closed intervals, is piecewise monotone (resp. piecewise strictly monotone, resp. piecewise linear) if there are finitely many points a0 < a1 < · · · < am in I such that I = [a0, am] and f is monotone (resp. strictly monotone, resp. affine) on [ai−1, ai] for i = 1, . . . ,m. We shall denote the class of these maps by M(I, J) (resp. S(I, J), resp. L(I, J)). We shall also denote the class of all continuous maps from I into J by C(I, J). In addition, the class of all closed intervals will be denoted by I, the set of all positive integers by Z+ and the set of all continuous n-th roots of f by n √ f . For an interval J we shall denote its boundary (i.e. the set of its endpoints) by bdJ . Clearly, any iterate of a map fromM(I, I), S(I, I) or L(I, I) has to belong to the same class. Therefore if we look for piecewise monotone (resp. piecewise strictly monotone, resp. piecewise linear) roots of f then we can restrict our attention to piecewise monotone (resp. piecewise strictly monotone, resp. piecewise linear) f . We obtain the following results.