Three Dimensional Partition and Infinite Renormalization

We provide a detailed construction of the three dimensional partition for the Julia set of an infinitely renormalizable quadratic polynomial. We show that for an unbranched infinitely renormalizable quadratic polynomial having complex bounds, the three-dimensional partition determines points in the Julia set dynamically. Furthermore, we construct a partition in the parameter space about a subset consisting of infinitely renormalizabe points. We prove that this subset is dense on the boundary of the Mandelbrot set. We show that every point in this subset is an unbranched quadratic polynomial having complex bounds. So its Julia set is locally connected. Moreover, we prove that the Mandelbrot set is locally connected at every point in this subset.