幂律族中动态确定区间的递减族
Decreasing families of dynamically determined intervals in the power-law family
摘要：
本文研究了拟二次族$x\mapsto -|x|^α+a.$中映射的后临界轨道所限定的区间之比的增长率。临界阶数$α$是任意实数$α>1.$。参数$a$的取值范围仅限于区间$(1,a_α)$，其长度取决于临界阶数。我们证明了在每个幂律族中存在唯一的参数$p_α$对应于捏合序列$RLRRRLRC.$。随后，我们得到了在捏合序列$RLRL...$的情况下，关于所有区间的比率的单调性结果。这推广了\cite{P}的结果。
英文摘要：
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.
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