We explore geometric properties of the Mandelbrot set \({{\mathcal {M}}}\), and the corresponding Julia sets \({{\mathfrak {J}}}_c\), near the main cardioid. Namely, we establish that: (a) \({{\mathcal {M}}}\) is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; (b) The Julia sets \({{\mathfrak {J}}}_c\) are also locally connected and have positive area; (c) \({{\mathcal {M}}}\) is self-similar near Siegel parameters of periodic type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed by the authors jointly with N. Selinger in [DLS] as a global transcendental family. It is the first occasion when external rays and puzzles of limiting transcendental maps are applied to study the Polynomial dynamics.

我们探索主心形线附近的Mandelbrot 集\({{\mathcal {M}}}\)和相应的 Julia 集\({{\mathfrak {J}}}_c\)的几何性质。也就是说，我们建立： (a) \({{\mathcal {M}}}\)在有界卫星类型的某些无限可重整化参数c处局部连接，提供了此类的第一个例子； (b) Julia 集合\({{\mathfrak {J}}}_c\)也是局部连通的并且具有正面积； (c) \({{\mathcal {M}}}\)在周期型西格尔参数附近是自相似的。我们通过分析作者与 N. Selinger 在 [DLS] 中作为全局超越族共同构建的 pacman 重整化算子的不稳定流形来解决这些问题。这是首次应用外部射线和极限超越图谜题来研究多项式动力学。