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I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$.

I know this question is known for a number of special cases. For example, if the $c$ is on the boundary of the Mandelbrot set, it has dimension 2. The dimension for $c=0$ is obvious as well. Are there any other cases known exactly? If so, how are they found? I'd imagine there are a number of measurements using box-counting methods to approximate the dimension for various cases.

Also, have there been efforts to calculate the dimension of a Julia set for any polynomial or rational function $p : \Bbb{C} \to \Bbb{C}$?

Any knowledge of work done in this area or a place to start would be awesome.

EDIT: Googling the question led to a number of papers. These are the results I have found:

This paper gives the dimension of some set of points, although googling the word "biaccesible" only brings up that paper and references to it.

This one shows that Julia sets for $c$ arbitrarily close to the boundary of the Mandelbrot set have dimensions arbitrarily close to 2.

This one [pdf] gives a number of results:

  • The dimension of $J(f)$ is less than 2 if $f$ has no non-periodic recurrent critical points
  • The Julia set of rational $f$ is hyperbolic $\implies$ The Hausdorff dimension as a function of $c$ is continuous
  • Some other results that seem to require another paper

The introduction of this paper Says that for a rational $f$ we have yet to find a Julia set with positive area and doesn't contain the whole Riemann sphere. (Doesn't $z \mapsto z^2$ provide a counterexample? I'm not sure I understand this one) I also proves a result about the Julia set of a $\sin $ function.

A paper from Harvard [pdf] gives ways of calculating the dimension numerically, and proves that the Hausdorff dimension is continuous from the Feigenbaum point (is this the same one from bifurcation diagrams?) to 1/4.

I'm going to read through more and these ones more carefully. In the meantime any guidance would help.

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A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start? –  CBenni Jan 27 '13 at 20:04
I just found some too. I'll read through them and compile what I find into my question, but I'm still looking for someone who knows the question far better than me to sum up present knowledge. –  Sam DeHority Jan 27 '13 at 20:07
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is –  Sheldon L Jan 28 '13 at 20:08
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A. –  Sam DeHority Jan 28 '13 at 20:45

1 Answer 1

up vote 1 down vote accepted

See also

  • Saupe, Dietmar, Efficient computation of Julia sets and their fractal dimension. Phys. D 28 (1987), no. 3, 358–370. DOI: 10.1016/0167-2789(87)90024-8

  • Jenkinson, Oliver; Pollicott, Mark, Calculating Hausdorff dimensions of Julia sets and Kleinian limit sets. Amer. J. Math. 124 (2002), no. 3, 495–545. DOI 10.1353/ajm.2002.0015

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I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv? –  Sam DeHority Jan 31 '13 at 3:10
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf . –  lhf Jan 31 '13 at 9:47

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