Orbits of Darboux-like real functions

Abstract

We show that, with respect to the dynamics of iteration, Darbouxlike functions from ℝ to 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 ℝ to ℝ which is 'universal for orbits' in the sense that it possesses every orbit of every function from ℝ to ℝ up to an arbitrary small translation, and which has orbits asymptotic to any real sequence, (ii) there is a function f: ℝ → ℝ such that for every n ∈ ℕ, fn is almost continuous and the graph of fn is dense in ℝ2, in spite of the fact that all f-orbits are finite. To prove (i) we assume the Continuum Hypothesis.