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

2007/09/11 by John Taylor, Taylor, John
Mathematics · #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37E05

paper · pdf · doi:10.48550/arxiv.0709.1595

five (5) pages. one typo corrected

arxiv created 2007/09/28 · arxiv updated 2011/11/09

Abstract

In this note we consider a collection \calC of one parameter families of unimodal maps of [0,1]. Each family in the collection has the form \μf\ where μ∈ [0,1]. Denoting the kneading sequence of μf by K(μf), we will prove that for each member of \calC, the map μ↦ K(μf) is monotone. It then follows that for each member of \calC the map μ↦ h(μf) is monotone, where h(uf) is the topological entropy of μf. For interest, μf(x)=4μx(1-x) and μf(x)=μsin(πx) are shown to belong to \calC. This extends the work of Masato Tsujii [1].

Related

• A Solution to the Monotonicity Problem for Unimodal Families
• Monotonicity of entropy for unimodal real quadratic rational maps
• Monotonicity of entropy for real multimodal maps
• Computing the topological entropy of unimodal maps
• On Entropy and Monotonicity for Real Cubic Maps