New insights into the combinatorial structure of the the Mandelbrot set are given by ‘Correspondence’ and ‘Translation’ Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them. Stony Brook IMS Preprint #1997/14 October 1997 1. The conjectures by Lau and Schleicher Introduction. The detailed ...

The external argument theory of Douady and Hubbard allows us to know both the potential and the field-lines in the exterior of the Mandelbrot set. Nonetheless, there are no explicit formulae to operate with external arguments, and the external argument theory is difficult to apply. In this paper we introduce some tools in order to obtain formulae to operate with external arguments in the Mandel...

In recent literature, researchers have generated Julia sets and Mandelbrot sets in Mann and Ishikawa orbits that are examples of two-step and three-step feedback processes respectively. This paper presents further generalization of Julia and Mandelbrot sets for complex-valued polynomials such as quadratic, cubic and higher degree polynomials using a Noor orbit, which is a four-step iterative pr...

Dual numbers, split-quaternions, split-octonions, and other number systems with nilpotent spaces have received sporadic yet persistent interest, beginning from their roots in the 19th century, to more recent attention in connection with supersymmetry in physics. In this paper, a number system in the 2D plane is investigated, where the squares of its basis elements p and q each map into the coor...

Combinatorial search methods often exhibit a large variability in performance. We study the cost prooles of combinatorial search procedures. Our study reveals some intriguing properties of such cost prooles. The distributions are often characterized by very long tails or \heavy tails". We will show that these distributions are best characterized by a general class of distributions that have no ...

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which ref...

We study two variants of the notion of holes formed by planar simple random walk of time duration 2n and the areas associated with them. We prove in both cases that the number of holes of area greater than A(n), where {A(n)} is an increasing sequence, is, up to a logarithmic correction term, asymptotic to n · A(n) for a range of large holes, thus confirming an observation by Mandelbrot. A conse...

Gray codes have found many applications in engineering and mathematics. In this work we explain how to apply Gray codes to the determination of the control parameter and initial point of a chaotic orbit generated by the Mandelbrot map when the kneading sequence is known. Possible applications to the cryptanalysis of a certain type of cryptosystems based on Baptista’s algorithm are also discussed.

Using Monte-Carlo simulations, we estimate numerically disconnec-tion exponents for planar Brownian motions. These simulations tend to connrm conjectures by Duplantier and Mandelbrot.

Non-self-similar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to (ideal) decision procedure, that is connected with undecidability. A halting set of a universal Turing machine (UTM), the Mandelbrot set and a riddled basin are mainly investigated as non-self-similar sets with a decision procedure. By encoding a symbol sequence to ...