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For a set X and family of functions ${\\cal F} = \\{ f\\sb{i}\\} \\sb{i\\in I}$ indexed by a countable set I, each $f\\sb{i} : X \\to X,$ we consider all possible ways of forming the $n\\sp{th}$ iterate, for $n\\in {\\rm I\\!N}.$ We study the dynamics of multiple valued maps which arise from functions of the form $B\\sb{r}(x) = rx$ (mod 1) as maps of the interval. For these systems, we determine the structure of orbits and study their discrete time averages.","Made available in DSpace on 2011-05-07T12:40:25Z (GMT). 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