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

We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family $x\mapsto -|x|^\alpha +a.$Thecriticalorder$\alpha$isanarbitraryrealnumber$\alpha>1.$Therangeoftheparameter$a$isconfinedtoaninterval$(1,a_{\alpha})$ of length depending on the critical order. We prove that in every power-law family there is a unique parameter $p_{\alpha }$ corresponding to the kneading sequence $RLRRRLRC\mathrm{.}$ Subsequently, we obtain monotonicity results concerning ratios of all intervals labeled by infinite post-critical orbit in the case of the kneading sequence $RLRL\mathrm{.}\mathrm{.}\mathrm{.}$ This extends the results from \cite{P}, via refinement of the tools based on special properties of power-law mappings in non-euclidean metric.