Hausdorff Dimension of Julia Sets for Non-Renormalizable Quadratic Polynomials concerning the Local Connectivity of the Mandelbrot Set

Mathematical Applications of Science Fiction We present a study on the Hausdorff dimension of the Julia sets of non-renormalizable quadratic polynomials, focusing on their behavior as the parameter approaches the boundary of the Mandelbrot set. By constructing a sequence of Yoccoz puzzles and analyzing the associated geometric structures, we establish a condition for the local connectivity of the Mandelbrot set. The results show that when the geometric divergence of the system grows faster than linearly, the Hausdorff dimension of the corresponding Julia set is strictly less than two, which implies rigidity of the conformal structure. The approach applies Patterson–Sullivan theory to the limit set generated by the puzzle pieces, extending the classical results of Yoccoz, Shishikura, and Lyubich within a unified operator-theoretic framework.