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

TL;DR

This paper proves that for certain one-parameter families of unimodal maps, the kneading sequence and topological entropy vary monotonically with the parameter, extending previous results to new classes including specific quadratic and sinusoidal maps.

Contribution

It establishes the monotonicity of kneading sequences and topological entropy for a broad class of unimodal maps parameterized by a single variable, including specific quadratic and sinusoidal examples.

Findings

Monotonicity of kneading sequences with respect to the parameter.

Monotonicity of topological entropy with respect to the parameter.

Includes specific examples like quadratic and sinusoidal maps.

Abstract

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].