论文标题
二次家族中乘数的临界点的累积集
Accumulation set of critical points of the multipliers in the quadratic family
论文作者
论文摘要
一个参数$ c_0 \ in \ mathbb c $在二次多项式的家族中$ f_c（z）= z^2+c $是一个周期$ n $乘数的关键点，如果地图$ f_ {c_0} $具有周期性的$ n $，其乘数$ n $，其倍增范围$ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c。 $ C = C_0 $。我们研究了乘数的累积集$ \数学x $，为$ n \ to \ infty $。这项研究补充了作者先前获得的乘数的临界点的等分分配结果。特别是，在当前的论文中，我们证明了累积集$ \ Mathcal x $有界，路径连接并包含Mandelbrot集作为适当的子集。我们还为累加集合$ \ MATHCAL X $包含的Mandelbrot集外的参数提供了必要且充分的条件，并证明该条件已满足一组开放的参数。我们的状况与定义曼德布罗特集合的条件之一相似。作为一个应用程序，我们得到将$ c $发送到$ f_c $的hausdorff尺寸的功能，在累积集$ \ mathcal x $之外没有关键点。
A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We study the accumulation set $\mathcal X$ of the critical points of the multipliers, as $n\to\infty$. This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper we prove that the accumulation set $\mathcal X$ is bounded, path connected and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set $\mathcal X$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends $c$ to the Hausdorff dimension of $f_c$, does not have critical points outside of the accumulation set $\mathcal X$.
