Mandelbrot set for fractal $n$-gons and zeros of power series

A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each $M_n$ can be called a binary matrix) and have eigenvalues whose negatives $-λ= c$ give periodic orbits under the Mandelbrot iteration, namely $z_k = z_{k-1}^2+c$ with $z_0=0$, and are thus contained in the Mandelbrot set. By the Perron--Frobenius theorem, the matrices $M_n$ have a dominant real positive eigenvalue, which we call $ρ_n$. This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and si…

Projections of Mandelbrot percolation in higher dimensions

We consider fractal percolation (or Mandelbrot percolation) which is one of the most well studied example of random Cantor sets. Rams and the first author studied the projections (orthogonal, radial and co-radial) of fractal percolation sets on the plane. We extend their results to higher dimension.

黎曼ζ函数的分形地理

Fractal Geography of the Riemann Zeta Function

二次Mandelbrot集被称为数学中最复杂、最美丽的对象，而黎曼ζ函数则被誉为最复杂、最神秘的函数。在这里，我们阐明了Zeta的Mandelbrot集和朱利亚集的频谱，以挖掘其混乱和分形的地理，将这两个极端结合成一个无畏的旅程，进入复杂的函数空间的最深处。

New Approximations for the Area of the Mandelbrot Set

Due to its fractal nature, much about the area of the Mandelbrot set $M$ remains to be understood. While a series formula has been derived by Ewing and Schober to calculate the area of $M$ by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing algorithm and for the 2-adic valuation of the series coefficients in terms of the sum-of-digits function.

Higher Order Mandelbrot Sets and their Varying Shapes

Mandelbrot set arose from the pioneering work of French mathematician Gaston Julia in the field of complex dynamics at the beginning of the 20th century. French-American mathematician Benoit Mandelbrot used computers to calculate iterations of complex polynomials of second order and displayed intricate images of fractal geometry. While studying fundamental properties of the Mandelbrot set, little attention has been paid to study the relationship between the degree of the generating polynomials and the shape of the main body of the Mandelbrot set. This paper extends the work from generating pol…

Geometrical representation of real and reactive powers of load demand by orbit diagrams in the Mandelbrot set

This paper presents the geometrical representation of the load demand by using orbits diagrams in the Mandelbrot set, to identify changing behaviors during a day period of the real and reactive powers. To perform this, different power combinations were used to represent the fractal diagrams with an algorithm that considers the mathematical model of Mandelbrot set and orbits diagrams. A qualitative analysis of the orbits is performed to identify the fractal graphic patterns with respect to the real and reactive power consumptions. The results show repetitive graphic patterns in the fractal spac…