Rigorous bounds for polynomial Julia sets

Luiz Henrique de  Figueiredo; Diego  Nehab; Jorge  Stolfi; João Batista S. de  Oliveira

doi:10.3934/jcd.2016006

Article Contents

2016, Volume 3, Issue 2: 113-137. Doi: 10.3934/jcd.2016006

This issue Previous Article Next Article Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator

Rigorous bounds for polynomial Julia sets

1.
IMPA, Rio de Janeiro
2.
Instituto de Computacão, UNICAMP, Campinas
3.
Faculdade de Informática, PUC-RS, Porto Alegre

Received: March 2016

Revised: August 2016

Published: June 2016

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Abstract

We present an algorithm for computing images of polynomial Julia sets that are reliable in the sense that they carry mathematical guarantees against sampling artifacts and rounding errors in floating-point arithmetic. We combine cell mapping based on interval arithmetic with label propagation in graphs to avoid function iteration and rounding errors. As a result, our algorithm avoids point sampling and can reliably classify entire rectangles in the complex plane as being on either side of the Julia set. The union of the rectangles that cannot be so classified is guaranteed to contain the Julia set. Our algorithm computes a refinable quadtree decomposition of the complex plane adapted to the Julia set which can be used for rendering and for approximating geometric properties such as the area of the filled Julia set and the fractal dimension of the Julia set.

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Mathematics Subject Classification: 37-04, 37F50, 37F10, 37M99, 65P99.

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