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Assignment 10: Fractal Dimension of Julia Sets

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import mmf_setup; mmf_setup.nbinit()
import os
from pathlib import Path
import logging
logging.getLogger("matplotlib").setLevel(logging.CRITICAL)
logging.getLogger('numba').setLevel(logging.CRITICAL)
logging.getLogger('OMP').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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  • Choose "Trust Notebook" from the "File" menu.
  • Re-execute this cell.
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Assignment 10: Fractal Dimension of Julia Sets#

What is the fractal dimension of the Julia set \(J_{N g(z)}\) corresponding to Newton’s method for finding the roots of \(g(z) = z^3 - 1\)? Use your previous code examples and assignments to estimate this with uncertainties.

Advanced/HPC: Color the complex plane \(c\in \mathbb{C}\) with the function \(\dim_{H}(J_c)\) where \(J_c\) is the standard Julia set corresponding to the iteration \(z\mapsto z^2 + c\).

Note

Recall that the Julia set consists of the boundary points. In the Newton case, these are the points that are not attracted to any of the roots (i.e. that take infinitely long to converge). In the standard case, these points are the boundary \(J_c = \partial K_c\) of the filled-in Julia set \(K_c\) which contains all points that do not escape.)

References#

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