We show small Mandelbrot sets are dense in the bifurcation locus for any holomorphic family of rational maps.

John Realpe-Gomez, ∗ Bartosz Szczesny, 3, † Luca Dall’Asta, 4, ‡ and Tobias Galla § Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom The Abdus Sal...

The aim of this article is to give some estimations of the Mandelbrot set. We also compute how big is the radius of a ball contained entirely into the Mandelbrot set.

We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. In the early 1990’s, Devaney asked the question: how can you tell where in the Mandelbrot set a given external ray lands, without having Adrien Douady at your side? We provide an answer t...

We introduce shrubs in this paper in order to present a first approach to studying the structure of the Mandelbrot set. Primary, secondary ... and N -ary shrubs are analyzed. We have experimentally obtained formulae to calculate the periods of the hyperbolic components representatives of the structural branches, and the preperiods and periods of both the nodes -where structural branches emanate...

The choice of the memory factor a tunes the intensity of memory: from the ahistoric model with a ! 0 to equally weighted records or full memory with a ! 1. Note in equation (1) that only an additional number HwL needs to be stored to implement this kind of memory, to be referred to as a-memory. Limited trailing memory would remember only the last t time steps. Thus mT ! m Hzß, ... , zTL, with ß...