On convex iterative roots of non-monotonic mappings

Let I be an interval. We consider the non-monotonic convex self-mappings f:I→ I such that f^2 is convex. They have the property that all iterates f^n are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every n≥ 2 there exists a convex iterative root f^1/2^n of order 2^n and the sequence {f^1/2^n} satisfies the condition of compatibility, that is f^1/2^n∘ f^1/2^n= f^1/2^n-1. A function f is said to be flowly convex if it possesses a convex semi-flow of f , that is a family of convex functions {f^t,t>0} such that f^t∘ f^s=f^t+s, t,s >0 and f^1=f . We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions.