A mapping f: R → R is called a total expansion if (Formula presented.) and (Formula presented.) for all a < b ∈ R; here fn stands for the nth iteration of f. We prove that there exists a smooth total expansion f: R → R such that one of its orbits is a given countable everywhere dense set. We also prove that, for each total expansion f: R → R, there exists a compact set K ⊂ R, referred to as an f-universal compact set, such that the sequence fn(K) is dense in the space Comp(R) of all nonempty compact subsets of R with the Hausdorff metric.
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