Periodic orbits of piecewise monotone maps

Abstract

Much is known about periodic orbits in dynamical systems of continuous interval maps. Of note is the theorem of Sharkovsky. In 1964 he proved that, for a continuous map $f$ on $\mathbb{R}$, the existence of periodic orbits of certain periods force the existence of periodic orbits of certain other periods. Unfortunately there is currently no analogue of this theorem for maps of $\mathbb{R}$ which are not continuous. Here we consider discontinuous interval maps of a particular variety, namely piecewise monotone interval maps. We observe how the presence of a given periodic orbit forces other periodic orbits, as well as the direct analogue of Sharkovsky's theorem in special families of piecewise monotone maps. We conclude by investigating the entropy of piecewise linear maps. Among particular one parameter families of piecewise linear maps, entropy remains constant even as the parameter varies. We provide a simple geometric explanation of this phenomenon known as entropy locking.