Orbit distributions of iterated function systems with finitely many forms

Let F={fi}i∈IF={fi}i∈I be a finite family of measure preserving self maps on a complete measure space (X,Σ,μ)(X,Σ,μ), indexed by the set II. For a sequence α=a1α=a1, a2a2, ……, where ai∈Iai∈I the nn-fold composition with respect to αα is Fαn=fan∘Fαn−1. When the nn-fold compositions from the family FF take finitely many forms, the discrete time distribution of the orbit of Fαk(x0) is a weighted average of the discrete time distributions of the orbits of the finite forms at the point x0x0 for μμ-almost all x0x0 and for almost all sequences αα. The weighted average is arrived at by showing that an independence condition holds through an application of the strong law of large numbers to a subsequence of the Rademacher functions. When the discrete time distributions of the finite forms are identical for μμ-almost all x0∈Xx0∈X the weighted sum of the discrete time distributions reduces to the single valued distribution for any one of the finite forms.