Mathematical Definition of Randomness

We define randomness for transformations via four criteria: maximal topological entropy,maximal Kolmogorov–Sinai entropy, statistical equidistribution, and the Martin–Löf criterion.These criteria emerged independently from four mathematical research programs between the1930s and 1960s. We prove that the generalized shift on Taylor’s Cantor space satisfies all fourcriteria. The Bernoulli shift and logistic map also satisfy this definition