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Let F={f"i}"i"@?"I be a finite family of measure preserving self maps on a complete measure space (X,@S,@m), indexed by the set I. For a sequence @a=a"1, a"2, ..., where a"i@?I the n-fold composition with respect to @a is F"@a^n=f"a"""n@?F"@a^n^-^1. When the n-fold compositions from the family F take finitely many forms, the discrete time distribution of the orbit of F"@a^k(x"0) is a weighted average of the discrete time distributions of the orbits of the finite forms at the point x"0 for @m-almost all x"0 and for almost all sequences @a. 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 @m-almost all x"0@?X the weighted sum of the discrete time distributions reduces to the single valued distribution for any one of the finite forms. |
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