Algorithmic randomness of continuous functions

George Barmpalias, Paul Brodhead, Douglas A. Cenzer, Jeffrey B. Remmel, Rebecca Weber

Abstract: We investigate notions of randomness in the space \({{\mathcal C}(2^{\mathbb N})}\) of continuous functions on \({2^{\mathbb N}}\). A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random \({\Delta^0_2}\) continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any \({y \in 2^{\mathbb N}}\), there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.