On the dimension of a part of the Mandelbrot set

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets

It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c ∈ ∂M , the Julia set of z 7→ z + c also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.

Topological Entropy of Quadratic Polynomials and Dimension of Sections of the Mandelbrot Set

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...

Percolation Dimension of Fractals

“Percolation dimension” is introduced in this note. It characterizes certain fractals and its definition is based on the Hausdorff dimension. It is shown that percolation dimension and “boundary dimension” are in a sense independent from the Hausdorff dimension and, therefore, provide an additional tool for classification of fractals. A unifying concept of fractal was introduced by Mandelbrot t...

Mathematical methods used in monofractal and multifractal analysis for the processing of biological and medical data and images

An object is self-similar if it can be decomposed into smaller copies of itself (the structure of the whole is contained in its parts) (Mandelbrot 1982; Falconer 2003). Not all irregular shapes found in nature are necessarily fractals (Losa 2009). Monofractal objects are mainly characterized by four properties: a) irregularity of the shape; b) self-similarity of the structure; c) non-integer or...