On the dynamics of total expansions of the real line

Abstract

A mapping f: R → R is called a total expansion if $\max (f^n ([a,b]))\mathop \to \limits_{n \to \infty } + \infty$ and $\min (f^n ([a,b]))\mathop \to \limits_{n \to \infty } - \infty$ for all a < b ∈ R; here f ⁿ 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 f ⁿ(K) is dense in the space Comp(R) of all nonempty compact subsets of R with the Hausdorff metric.