Efficient computations of Julia sets and their fractal dimension
type: Item
instance of: scholarly article
title: Efficient computations of Julia sets and their fractal dimension
zbMATH Open document ID: 0622.58015
DOI: 10.1016/0167-2789(87)90024-8
author: Dietmar Saupe
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: The computation of the fractal dimension is straightforward using the box-counting method. However, this approach may require very long computation times. If the Julia set is the connected common boundary of two or more basins of attraction, then a recursive version of the box- counting method can be made storage- and time-efficient. The method is also suitable for the computation of the Julia sets. We apply the method to verify a result of D. Ruelle regarding the dimension of Julia sets of \(R(z)=z^ 2+c\) for small \(c\in {\mathbb{C}}\), to Newton's method for complex polynomials of degree 3 and to a sequence of Julia sets from the renormalization transformation for hierarchical lattices. We also discuss the computation of Julia sets and their information dimension by the inverse iteration method. In all examples tested we find that the information dimension is less than the fractal dimension.
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 4009373
zbMATH Keywords: fractal dimension
zbMATH Keywords: box-counting method
zbMATH Keywords: Julia set
zbMATH Keywords: basins of attraction
Wikidata QID: Q121438612
MaRDI profile type: Publication
full work available at URL: 0167-2789(87)90024-8
OpenAlex ID: W2004423816
cites work: Complex analytic dynamics on the Riemann sphere
cites work: On the iteration of a rational function: Computer experiments with Newton's method
cites work: Q5186444
cites work: Q3322903
cites work: Q4740120
cites work: Cayley's problem and Julia sets
cites work: Q3736578
cites work: Repellers for real analytic maps
cites work: Five turbulent problems
description: scientific article; zbMATH DE number 4009373