We consider a map called a double rotation, which is composed of two rotations on a circle. Specifically, a double rotation is a map on the interval [0, 1) that maps x ∈ [0, c) to {x + α}, and x ∈ [c, 1) to {x + β}. Although double rotations are discontinuous and non-invertible in general, we show that almost every double rotation can be reduced to a simple rotation, and the set of parameters s...

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each can be called a binary matrix) eigenvalues whose negatives $-\lambda = c$ give periodic orbits under the iteration, namely $z_k z_{k-1}^2+c$ with $z_0=0$, are thus contained in set. By Perron--Frobenius theorem, dominant real positive eigenvalue, wh...

D = lim log(.iV(A,c)) (1) E-o log1 where .Af(A, e) denotes the smallest number of closed balls of radius > 0 needed to cover the set A. The aim of this paper is to introduce a new fast algorithm for computing the fractal dimension of binary images according to (1). Consider a binary image that is represented as a matrix of binary pixels stored in a rectangular array I(x, y). Let this image deno...

A lot of formal and informal recreational study took place in the fields of Meromorphic Maps, since Mandelbrot popularized the map z ← z + c. An immediate generalization of the Mandelbrot z ← z + c also known as the Multibrot family were also studied. In the current paper, general truncated polynomial maps of the form z ← ∑ p≥2 apx p + c are studied. Two fundamental properties of these polynomi...

Gene promoter networks (GPNs) are systems-level representations of the base pair-sharing relationships (graph edges) among promoters (graph nodes). It has been shown in the bacterium E. coli that these networks can contain a fractal nucleus of strong associations suggesting a self-organizing complexity. Here I report results of twenty seven in silico simulations for a diffusion limited aggregat...