Common periodic trajectories of interval maps

Abstract.

We prove that for any continuous piecewise monotone or smooth interval map f and any subset ${\mathbb{M}}$ of the set of periods of periodic trajectories of f, there is another map $\tilde{f}$ such that the set of periods of periodic trajectories common for f and $\tilde{f}$, which is denoted by $p(f, \tilde{f})$, coincides with ${\mathbb{M}}$. At the same time, for each integer $m \geq 0$, there exists a continuous map f such that $2^m \in p(f, \tilde{f})$ for any map $\tilde{f}$ if $p(f, \tilde{f})$ is an infinite set.