Decreasing families of dynamically determined intervals in the power-law family

2007/04/11 by Waldemar Paluba, Paluba, Waldemar
Mathematics · #37D05 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37D05

paper · pdf · doi:10.48550/arxiv.0704.1385

arxiv created 2007/04/11 · arxiv updated 2009/12/01

Abstract

We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family x↦ -|x|^α+a. The critical order α is an arbitrary real number α>1. The range of the parameter a is confined to an interval (1,a_α) of length depending on the critical order. We prove that in every power-law family there is a unique parameter p_α corresponding to the kneading sequence RLRRRLRC. Subsequently, we obtain monotonicity results concerning ratios of all intervals labeled by infinite post-critical orbit in the case of the kneading sequence RLRL... This extends the results from \citeP, via refinement of the tools based on special properties of power-law mappings in non-euclidean metric.

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