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

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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,aα) of length depending on the critical order. We prove that in every power-law family there is a unique parameter pα 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 P, via refinement of the tools based on special properties of power-law mappings in non-euclidean metric.