A Generalized Mandelbrot Set Based On Distance Ratio

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

On the spectra of reduced distance matrix of the generalized Bethe trees

Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.

Combinatorics of distance doubling maps ∗

We study the combinatorics of distance doubling maps on the circle R/Z with prototypes h(β) = 2β mod 1 and h(β) = −2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where iterates f◦n of a distance doubling map f provide ‘distance doubling behavior’. The results include well-known statements for h relat...

Is the Mandelbrot set computable?

We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two–sided distance function ...

A ~e on some empirical properties of the Mandelbrot Set

In this note we inspect the kind of patterns produced by the quadratic iteration used to generate the Mandelbrot set. A very simple relation involving the Fibonacci numbers can be found between the relative positions of the disks which form the Mandelbrot set and the number of attractors of the iteration. Using this insight the concept of escape time can be generalized to points inside and not ...