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

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

2007-09-11

arxiv.org/abs/0709.1595v3

Mathematics

Dynamical Systems

five (5) pages. one typo corrected

Scientific paper

In this note we consider a collection $\cal{C}$ 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 $\cal{C}$, the map $\mu\mapsto K(\mu f)$ is monotone. It then follows that for each member of $\cal{C}$ the map $\mu\mapsto h(\mu f)$ is monotone, where $h(\u{f})$ 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 $\cal{C}.$ This extends the work of Masato Tsujii [1].

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