The Borel structure of iterates of continuous functions

Abstract

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f ∈C[0,1] mapping [0,1] into itself. If f ∈C, f k will denote the kth iterate of f and we put C k = { f k : f ∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I , we let C( f ) denote the set of points at which f is locally constant, i.e. We let N denote the set of positive integers and N N denote the Baire space of sequences of positive integers.