When will a One Parameter Family of Unimodal Maps Produce Finite Limit Cycles Monotonically with the Parameter?

Abstract

In this note we consider a collection $\mathcal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\{\mu f\}$ where $\mu \in [0,1].$ Denoting the kneading sequence of $\mu f$ by $K(\mu f)$, we will prove that for each member of $\mathcal{C}$, the map $\mu \mapsto K(\mu f)$ is monotone. It then follows that for each member of $\mathcal{C}$ the map $\mu \mapsto h(\mu f)$ is monotone, where $h(uf)$ is the topological entropy of $\mu f.$ For interest, $\mu f(x)=4\mu x(1-x)$ and $\mu f(x)=\mu \sin (\pi x)$ are shown to belong to $\mathcal{C}.$ This extends the work of Masato Tsujii [1].

arXiv commentfive (5) pages. one typo corrected