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.