A Chaotic Continuous Map Generates All Probability Distributions

Let ƒ be a continuous map of the compact unit interval I = [0, 1], such that ƒ2, the second iterate of ƒ, is topologically transitive in I. If for some x and y in I and any t in I there exists lim(1/n) # {i ≤ n; |ƒi(x) − ƒi(y)| < t} for n → ∞, denote it by φxy(t). In the paper we consider the class F(ƒ) if all φxy. The main results are that F(ƒ) is convex and pointwise closed. Using this we show that F(ƒ) is always bigger than the class D(ƒ) of probability distributions generated analogously by single trajectories (and corresponding to the class of probability invariant measures of ƒ), and prove that there are universal generators of probability distributions, i.e., maps ƒ such that F(ƒ) is the class M of all non-decreasing functions I ⇒ I (contrary to this, D(ƒ) ⊃ M for no ƒ). These results can be extended to more general continuous maps. One of the possible applications is to use the size of F(ƒ) as a measure of the degree of chaos of ƒ.