# Delta Hausdorff Dimension

Robert P. Munafo, 2010 Sep 7.

The box-counting dimension for a given neighborhood of a point, which is considered to be a term in a series, the limit of which would be the Hausdorff dimension.

It gives a rough qualitative estimate of the "apparent" Hausdorff dimension of an image at a given pixel resolution, and is computed by the following formula:

D_{dh} = log_{2}(POP_{2s} / POP_{s})

Here D_{dh} represents delta Hausdorff dimension, POP_{2s} is the
population (pixel count) on a pixel grid of size 2s, and POP_{s}
is the population on a grid of size s. Both pixel grids cover the same
area (as expressed in real and imaginary coordinates) but the second
grid has twice as many pixels (sample-points).

If P is a point in the Mandelbrot Set, D_{dh} for a neighborhood
of P is constrained to be in the range containing the Hausdorff
dimensions of the Julia Sets for all points within the
neighborhood. There are points in the Mandelbrot Set for which the
Julia Set has a Hausdorff dimension "arbitrarily close" to 2.0. Since
all neighborhoods of P contain embedded copies of the Mandelbrot
Set, the upper bound of the range for any neighborhood is 2.0. As you
zoom in, the neighborhood gets smaller and the lower bound of the
range can go up. However the lower bound does not always go up. Also,
the D_{dh} of the neighborhood can go up or down as you zoom in.

This is all sort of abstract, so I'll give some examples. To appreciate what's going on, compare the images to the numbers below. The number gives the fractal dimension of the image by the box-counting method; it can be anything from 1.00 to 2.00. The images were created using the distance estimator method.

In the first example, we're zooming in to the tip of a filament and
D_{dh} goes down and reaches an asymptote quickly:

coordinates | ||||

0.00000 +1.00000i |
radius 0.3 D_{dh} = 1.540 |
radius 0.1 D_{dh} = 1.349 |
radius 0.03 D_{dh} = 1.255 |
radius 0.01 D_{dh} = 1.238 |

radius 0.003 D_{dh} = 1.278 |
radius 0.001 D_{dh} = 1.248 |
radius 0.0003 D_{dh} = 1.278 |
radius 0.0001 D_{dh} = 1.257 |

In the second example D_{dh} goes up steadily as we zoom in, which
is the most common behavior for most points that people usually zoom
in to:

coordinates | ||||

-1.74856 +0.00075i |
radius 0.3 D_{dh} = 1.275 |
radius 0.1 D_{dh} = 1.389 |
radius 0.03 D_{dh} = 1.547 |
radius 0.01 D_{dh} = 1.548 |

radius 0.003 D_{dh} = 1.594 |
radius 0.001 D_{dh} = 1.663 |
radius 0.0003 D_{dh} = 1.706 |
radius 0.0001 D_{dh} = 1.765 |

In the third example D_{dh} goes up and then goes back down:

coordinates | |||||

-1.74831 | +0.00046i |
radius 0.3 D_{dh} = 1.361 |
radius 0.1 D_{dh} = 1.400 |
radius 0.03 D_{dh} = 1.568 |
radius 0.01 D_{dh} = 1.602 |

radius 0.003 D_{dh} = 1.583 |
radius 0.001 D_{dh} = 1.602 |
radius 0.0003 D_{dh} = 1.357 |
radius 0.0001 D_{dh} = 1.382 |

It can occur that D_{dh} goes up and down several or many times, but
you have to be willing to zoom in a really long way to see this
happen. One example is the point

-1.769 110 375 463 767 385 + 0.009 020 388 228 023 440 i

if you zoom in "all the way" on these coordinates you see this:

*the Munafo minibrot*

References

The Wikipedia page on box-counting dimension is here: Minkowski-Bouligand dimension

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jul 03. s.27