Hyperbolic components and Misiurewicz points of the chaotic region of the Mandelbrot set are located in what we call shrubs. Each shrub has two well-differentiated parts. The first one, shrub0; corresponds to the main branch of the shrub. The second one, shrubr; corresponds to all the other branches of the shrub. In this paper, we have focused on the ordering of the hyperbolic components and Mi...

A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy. Indeed, we prove an “entropy formula” relating the entropy of a polynomial...

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Mandelbrot Maps is a fractal explorer for the Mandelbrot Set and the Julia Sets allowing for their real time exploration. The fractal explorer also provides mechanisms for studying two important theorems that relate the two sets together Tan Lei’s similarity theorem and Fatou’s connectedness theorem. The tool has been used to show case how parallelisation techniques can be used to obtain better...

Using holomorphic surgery techniques, we construct a homeomorphism between a neighborhood of any limb without root point of the Mandelbrot set and a neighborhood of any other of equal denominator, in such a way that the limbs are mapped to each other. On the limbs, the homeomorphism coincides with that constructed in “Homeomorphisms between limbs of the Mandelbrot set” (J. Geom. Anal. 9 (1999),...

Let f be a rational function, which has k n-cycles under iteration. By using the symmetry of the underlying equation of degree k ·n, it is reduced to equations of degree k and n. This is explained in terms of Galois theory. The 3and 4-cycles of fc(z) = z2 + c are obtained explicitly. This yields the corresponding multiplier, which maps hyperbolic components of the Mandelbrot set conformally ont...