We construct a subset of the Mandelbrot set which is dense on the boundary of the Mandelbrot set and which consists of only infinitely renormalizable points such that the Mandelbrot set is locally connected at every point of this subset. We prove the local connectivity by finding bases of connected neighborhoods directly.

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The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number π while exploring a seemingly unrelated property of the Mandelbrot set. Boll’s Þnding is easy to describe and understand, and yet it is not widely known — possibly because the result has not been rigorously shown. The purpose of this paper is to prese...

This is a continuation of the work Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of a parabolic bifurcation of the Mandelbrot-like sets in the parameter space. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we obtain a parabolic Mandelbrot-like set contained in the family of maps with a x...

We iterate the Weierstrass elliptic ℘ function in order to understand the dependence of the dynamics on the underlying period lattice L. We focus on square lattices and use the holomorphic dependence on the classical invariants (g2, g3) = (g2, 0) to show that in parameter space (g2-space) one sees both quadratic-like attracting orbit behavior and pre-pole dynamics. In the case of pre-pole param...

Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated structure, and how the external rays lan...

The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can...

On subsets of the Mandelbrot set, EM ⊂M, homeomorphisms are constructed by quasi-conformal surgery. When the dynamics of quadratic polynomials is changed piecewise by a combinatorial construction, a general theorem yields the corresponding homeomorphism h : EM → EM in the parameter plane. Each h has two fixed points in EM , and a countable family of mutually homeomorphic fundamental domains. Po...