Decreasing families of dynamically determined intervals in the power-law family
Abstract
Description
We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family $x\mapsto -|x|^α+a.$ 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 \cite{P}, via refinement of the tools based on special properties of power-law mappings in non-euclidean metric.