A Solution to the Monotonicity Problem for Unimodal Families

TL;DR

This paper proves that for a certain class of unimodal map families, the kneading sequence and topological entropy vary monotonically with the parameter, providing insights into their dynamic behavior.

Contribution

It establishes the monotonicity of kneading sequences and topological entropy for a broad class of unimodal families, including specific examples.

Findings

Kneading sequence u->K(uf) is monotone for all families in the class.

Topological entropy u->h(uf) is monotone for these families.

Examples include the logistic map and sine family.

Abstract

In this note we consider a collection C of one parameter families of unimodal maps of [0,1]. Each family in the collection has the form uf where u is in [0,1]. Denoting the kneading sequence of uf by K(uf), we will prove that for each member of C, the map u->K(uf) is monotone. It then follows that for each member of C the map u -> h(uf) is monotone, where h(uf) is the topological entropy of uf. For interest, uf(x)=4ux(1-x) and uf(x)=usin(pi x) are shown to belong to C.