Orbits of Darboux-Like Real Functions

Abstract

We show that, with respect to the dynamics of iteration, Darboux-like functions from $\mathbb{R}$ to $\mathbb{R}$ can exhibit some strange properties which are impossible for continuous functions. To be precise, we show that (i) there is an extendable function from $\mathbb{R}$ to $\mathbb{R}$ which is `universal for orbits' in the sense that it possesses every orbit of every function from $\mathbb{R}$ to $\mathbb{R}$ up to an arbitrary small translation, and which has orbits asymptotic to any real sequence, (ii) there is a function $f\:mathbb{R}\to \mathbb{R}$ such that for every $n\in \mathbb{N}$, $f^n$ is almost continuous and the graph of $f^n$ is dense in $\mathbb{R}^2$, in spite of the fact that all $f$-orbits are finite. To prove (i) we assume the Continuum Hypothesis.