A Solution to the Monotonicity Problem for Unimodal Families

Details A Solution to the Monotonicity Problem for Unimodal Families A Solution to the Monotonicity Problem for Unimodal Families

2008-05-22

arxiv.org/abs/0805.3436v1

Mathematics

Dynamical Systems

Scientific paper

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.

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