Mathematics – Dynamical Systems
Scientific paper
2007-04-11
Mathematics
Dynamical Systems
Scientific paper
We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family $x\mapsto -|x|^\alpha +a.$ The critical order $\alpha$ is an arbitrary real number $\alpha>1.$ The range of the parameter $a$ is confined to an interval $(1,a_{\alpha})$ 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 $RLRRRLRC.$ 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.
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