Hausdorff Dimension of Arbitrary Julia SetHausdorff dimension of graphs of one-dimensional Brownian motionHausdorff dimension of the set of rational numbers within a certain interval?Area of filled Julia setMinkowski dimension behaviour of sets with positive finite Hausdorff measure.How to plot the Julia Set of $z-z^2$What is the value of $z$ for any Julia set? Does it influence the graphical result?How come Hausdorff dimension of Mandelbrot is lower than Lorenz attractor?Properties about Julia setsHausdorff Dimension of Julia set of $z^2+2$?bounds on dimension of Julia sets inside Mandelbrot set
What do you call a Matrix-like slowdown and camera movement effect?
Why don't electron-positron collisions release infinite energy?
How is this relation reflexive?
If Manufacturer spice model and Datasheet give different values which should I use?
A newer friend of my brother's gave him a load of baseball cards that are supposedly extremely valuable. Is this a scam?
Why did the Germans forbid the possession of pet pigeons in Rostov-on-Don in 1941?
Can I interfere when another PC is about to be attacked?
Validation accuracy vs Testing accuracy
Email Account under attack (really) - anything I can do?
How to type dʒ symbol (IPA) on Mac?
How to add power-LED to my small amplifier?
What are these boxed doors outside store fronts in New York?
whey we use polarized capacitor?
How is it possible for user's password to be changed after storage was encrypted? (on OS X, Android)
How long does it take to type this?
Why are only specific transaction types accepted into the mempool?
How to make payment on the internet without leaving a money trail?
When blogging recipes, how can I support both readers who want the narrative/journey and ones who want the printer-friendly recipe?
Is Social Media Science Fiction?
What Brexit solution does the DUP want?
Japan - Plan around max visa duration
Why was the small council so happy for Tyrion to become the Master of Coin?
How old can references or sources in a thesis be?
Infinite past with a beginning?
Hausdorff Dimension of Arbitrary Julia Set
Hausdorff dimension of graphs of one-dimensional Brownian motionHausdorff dimension of the set of rational numbers within a certain interval?Area of filled Julia setMinkowski dimension behaviour of sets with positive finite Hausdorff measure.How to plot the Julia Set of $z-z^2$What is the value of $z$ for any Julia set? Does it influence the graphical result?How come Hausdorff dimension of Mandelbrot is lower than Lorenz attractor?Properties about Julia setsHausdorff Dimension of Julia set of $z^2+2$?bounds on dimension of Julia sets inside Mandelbrot set
$begingroup$
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 $p : BbbC to BbbC$?
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.
complex-analysis reference-request fractals geometric-measure-theory open-problem
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
$endgroup$
add a comment |
$begingroup$
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 $p : BbbC to BbbC$?
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.
complex-analysis reference-request fractals geometric-measure-theory open-problem
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
$endgroup$
$begingroup$
A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start?
$endgroup$
– CBenni
Jan 27 '13 at 20:04
1
$begingroup$
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.
$endgroup$
– guest196883
Jan 27 '13 at 20:07
1
$begingroup$
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is
$endgroup$
– Sheldon L
Jan 28 '13 at 20:08
$begingroup$
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A.
$endgroup$
– guest196883
Jan 28 '13 at 20:45
add a comment |
$begingroup$
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 $p : BbbC to BbbC$?
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.
complex-analysis reference-request fractals geometric-measure-theory open-problem
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
$endgroup$
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 $p : BbbC to BbbC$?
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.
complex-analysis reference-request fractals geometric-measure-theory open-problem
complex-analysis reference-request fractals geometric-measure-theory open-problem
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
edited Dec 18 '14 at 5:34
d80d2729a352b1366139fc119d3345
5,18022351
5,18022351
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
asked Jan 27 '13 at 19:46
guest196883guest196883
5,0591241
5,0591241
$begingroup$
A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start?
$endgroup$
– CBenni
Jan 27 '13 at 20:04
1
$begingroup$
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.
$endgroup$
– guest196883
Jan 27 '13 at 20:07
1
$begingroup$
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is
$endgroup$
– Sheldon L
Jan 28 '13 at 20:08
$begingroup$
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A.
$endgroup$
– guest196883
Jan 28 '13 at 20:45
add a comment |
$begingroup$
A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start?
$endgroup$
– CBenni
Jan 27 '13 at 20:04
1
$begingroup$
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.
$endgroup$
– guest196883
Jan 27 '13 at 20:07
1
$begingroup$
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is
$endgroup$
– Sheldon L
Jan 28 '13 at 20:08
$begingroup$
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A.
$endgroup$
– guest196883
Jan 28 '13 at 20:45
$begingroup$
A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start?
$endgroup$
– CBenni
Jan 27 '13 at 20:04
$begingroup$
A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start?
$endgroup$
– CBenni
Jan 27 '13 at 20:04
1
1
$begingroup$
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.
$endgroup$
– guest196883
Jan 27 '13 at 20:07
$begingroup$
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.
$endgroup$
– guest196883
Jan 27 '13 at 20:07
1
1
$begingroup$
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is
$endgroup$
– Sheldon L
Jan 28 '13 at 20:08
$begingroup$
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is
$endgroup$
– Sheldon L
Jan 28 '13 at 20:08
$begingroup$
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A.
$endgroup$
– guest196883
Jan 28 '13 at 20:45
$begingroup$
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A.
$endgroup$
– guest196883
Jan 28 '13 at 20:45
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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)90024-8Jenkinson, 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
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
$endgroup$
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
add a comment |
$begingroup$
There is NO exact solution for the Hausdorff dimension of Julia sets for $z^2+c$ for general $c$. There are numerical ways to compute approximately (which I'm not vary familiar with), but the exact value is known only for very exceptional case.
Also, it is true that the Hausdorff dimension is equal to two for GENERIC c in the boundary of the Mandelbrot set: see Mitsuhiro Shishikura, "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets".
Moreover, it is known that the area of the Julia set can be positive for some $c$: see Xavier Buff and Arnaud Chéritat, "Quadratic Julia sets with positive area" or by Artur Avila and Mikhail Lyubich, "Lebesgue measure of Feigenbaum Julia sets"
(Note that the McMullen's paper on positive area is not for rational maps, but for sine family $f(z) = lambda sin z$ (and similar ones), which is a family of transcendental entire maps).
However, at the same time, there are many cases that the Hausdorff dimension is strictly smaller than two. For example, as already mentioned, if the critical points are non-recurrent, the Hausdorff dimension is strictly less than two: see Mariusz Urbański, "Rational functions with no recurrent critical points".
And there are many $c$ in the boundary of the Mandelbrot set satisfying this property.
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
answered Mar 22 at 8:44
inyoinyo
111
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f288284%2fhausdorff-dimension-of-arbitrary-julia-set%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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)90024-8Jenkinson, 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
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
$endgroup$
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
add a comment |
$begingroup$
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)90024-8Jenkinson, 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
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
$endgroup$
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
add a comment |
$begingroup$
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)90024-8Jenkinson, 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
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
$endgroup$
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)90024-8Jenkinson, 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
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
answered Jan 28 '13 at 23:31
lhflhf
167k11172404
167k11172404
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
add a comment |
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
I can't get access to the first one, and a google seached turned up nothing. Is it on arXiv?
$endgroup$
– guest196883
Jan 31 '13 at 3:10
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
$begingroup$
@DoctorBatmanGod, Google found copies of both papers: inf.uni-konstanz.de/gk/pubsys/publishedFiles/Saupe87.pdf and maths.qmul.ac.uk/~omj/schottky11.pdf .
$endgroup$
– lhf
Jan 31 '13 at 9:47
add a comment |
$begingroup$
There is NO exact solution for the Hausdorff dimension of Julia sets for $z^2+c$ for general $c$. There are numerical ways to compute approximately (which I'm not vary familiar with), but the exact value is known only for very exceptional case.
Also, it is true that the Hausdorff dimension is equal to two for GENERIC c in the boundary of the Mandelbrot set: see Mitsuhiro Shishikura, "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets".
Moreover, it is known that the area of the Julia set can be positive for some $c$: see Xavier Buff and Arnaud Chéritat, "Quadratic Julia sets with positive area" or by Artur Avila and Mikhail Lyubich, "Lebesgue measure of Feigenbaum Julia sets"
(Note that the McMullen's paper on positive area is not for rational maps, but for sine family $f(z) = lambda sin z$ (and similar ones), which is a family of transcendental entire maps).
However, at the same time, there are many cases that the Hausdorff dimension is strictly smaller than two. For example, as already mentioned, if the critical points are non-recurrent, the Hausdorff dimension is strictly less than two: see Mariusz Urbański, "Rational functions with no recurrent critical points".
And there are many $c$ in the boundary of the Mandelbrot set satisfying this property.
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
answered Mar 22 at 8:44
inyoinyo
111
$endgroup$
add a comment |
$begingroup$
There is NO exact solution for the Hausdorff dimension of Julia sets for $z^2+c$ for general $c$. There are numerical ways to compute approximately (which I'm not vary familiar with), but the exact value is known only for very exceptional case.
Also, it is true that the Hausdorff dimension is equal to two for GENERIC c in the boundary of the Mandelbrot set: see Mitsuhiro Shishikura, "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets".
Moreover, it is known that the area of the Julia set can be positive for some $c$: see Xavier Buff and Arnaud Chéritat, "Quadratic Julia sets with positive area" or by Artur Avila and Mikhail Lyubich, "Lebesgue measure of Feigenbaum Julia sets"
(Note that the McMullen's paper on positive area is not for rational maps, but for sine family $f(z) = lambda sin z$ (and similar ones), which is a family of transcendental entire maps).
However, at the same time, there are many cases that the Hausdorff dimension is strictly smaller than two. For example, as already mentioned, if the critical points are non-recurrent, the Hausdorff dimension is strictly less than two: see Mariusz Urbański, "Rational functions with no recurrent critical points".
And there are many $c$ in the boundary of the Mandelbrot set satisfying this property.
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
answered Mar 22 at 8:44
inyoinyo
111
$endgroup$
add a comment |
$begingroup$
There is NO exact solution for the Hausdorff dimension of Julia sets for $z^2+c$ for general $c$. There are numerical ways to compute approximately (which I'm not vary familiar with), but the exact value is known only for very exceptional case.
Also, it is true that the Hausdorff dimension is equal to two for GENERIC c in the boundary of the Mandelbrot set: see Mitsuhiro Shishikura, "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets".
Moreover, it is known that the area of the Julia set can be positive for some $c$: see Xavier Buff and Arnaud Chéritat, "Quadratic Julia sets with positive area" or by Artur Avila and Mikhail Lyubich, "Lebesgue measure of Feigenbaum Julia sets"
(Note that the McMullen's paper on positive area is not for rational maps, but for sine family $f(z) = lambda sin z$ (and similar ones), which is a family of transcendental entire maps).
However, at the same time, there are many cases that the Hausdorff dimension is strictly smaller than two. For example, as already mentioned, if the critical points are non-recurrent, the Hausdorff dimension is strictly less than two: see Mariusz Urbański, "Rational functions with no recurrent critical points".
And there are many $c$ in the boundary of the Mandelbrot set satisfying this property.
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
answered Mar 22 at 8:44
inyoinyo
111
$endgroup$
There is NO exact solution for the Hausdorff dimension of Julia sets for $z^2+c$ for general $c$. There are numerical ways to compute approximately (which I'm not vary familiar with), but the exact value is known only for very exceptional case.
Also, it is true that the Hausdorff dimension is equal to two for GENERIC c in the boundary of the Mandelbrot set: see Mitsuhiro Shishikura, "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets".
Moreover, it is known that the area of the Julia set can be positive for some $c$: see Xavier Buff and Arnaud Chéritat, "Quadratic Julia sets with positive area" or by Artur Avila and Mikhail Lyubich, "Lebesgue measure of Feigenbaum Julia sets"
(Note that the McMullen's paper on positive area is not for rational maps, but for sine family $f(z) = lambda sin z$ (and similar ones), which is a family of transcendental entire maps).
However, at the same time, there are many cases that the Hausdorff dimension is strictly smaller than two. For example, as already mentioned, if the critical points are non-recurrent, the Hausdorff dimension is strictly less than two: see Mariusz Urbański, "Rational functions with no recurrent critical points".
And there are many $c$ in the boundary of the Mandelbrot set satisfying this property.
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
answered Mar 22 at 8:44
inyoinyo
111
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
edited Mar 22 at 9:04
Daniele Tampieri
2,66221022
2,66221022
answered Mar 22 at 8:44
inyoinyo
111
answered Mar 22 at 8:44
inyoinyo
111
answered Mar 22 at 8:44
inyoinyo
111
111
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f288284%2fhausdorff-dimension-of-arbitrary-julia-set%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
A quick google search brought me a bunch of papers... I have not yet read through them, but wouldnt that be a start?
$endgroup$
– CBenni
Jan 27 '13 at 20:04
1
$begingroup$
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.
$endgroup$
– guest196883
Jan 27 '13 at 20:07
1
$begingroup$
I wonder what the Hausdorff dimension for C=0.25, the parabolic boundary case, is
$endgroup$
– Sheldon L
Jan 28 '13 at 20:08
$begingroup$
@SheldonL It is approximately 1.0812 according to the last harvard paper. It's on the first table in Appendix A.
$endgroup$
– guest196883
Jan 28 '13 at 20:45