Talk:Misiurewicz point

Rewritten
I have rewrite the article. --Adam majewski 10:59, 17 June 2007 (UTC)

definition
Hi what is the meaning and source of this statement :$$f_c^{(k-1)}(z_0) \neq f_c^{(k+n-1)}(z_0) \,$$ ?

Is it possible to compute number of Misiurewicz points for given preperiod and period for complex quadratic polynomial ?--Adam majewski 20:34, 22 July 2007 (UTC)


 * We want the iterations of the critical point, z0, to enter a periodic orbit at the kth iteration, but not before (otherwise we have a pre-period of less than k). So the condition


 * $$f_c^{(k)}(z_0) = f_c^{(k+n)}(z_0) \,$$


 * on its own is not sufficient. If the critical point enters a period n orbit with pre-period less than k, then the above condition is met - but in this case c will also satisfy


 * $$f_c^{(k-1)}(z_0) = f_c^{(k+n-1)}(z_0) \,$$


 * and this is the case that we want to exclude. Not sure about the answer to your second question - there will be 2k+n-1 solutions to the defining equation $$P_c^{(k)}(0) = P_c^{(k+n)}(0)$$, but not all of these will be Mk,n points because for some of them the critical point will have a pre-period shorter than k or a period that is a factor of n. Gandalf61 09:21, 23 July 2007 (UTC)

Ad 1. What is the source of this condition ? In papers or books, which I know I have never seen that.

Ad 2. Is it possible to made a formula or algorithm for computing such number ? I know solution of this problem for real case see pastor01.

3. ( new question) How to compute number of external rays landing on $$\ M_{n,k}$$ ?

4. ( new question) How to compute angles of external rays landing on $$\ M_{n,k}$$ ?

Adam majewski 14:57, 23 July 2007 (UTC)

Layman's explanation
Please forgive my ignorance, but could someone provide a simple exlanation for what this term means for the less mathematically-inclined? I came here from Mandelbrot Set (which article did give me a little more understanding of its topic) but all I see here is a bunch of formulas that don't mean anything to me, and some prose explanations that are so loaded with other terms (whose articles are likewise entirely cerebral) that I just get completely lost. There are actually a lot of math-related articles that have this same issue of being probably quite informative to experts, and technically accurate, but with such a high threshhold to understanding them that those without the technical background required can gain nothing whatsoever from them. P.S. Please respond on my talk page. Thanks,  D a n si m a n  ( talk | Contribs ) 14:21, 13 March 2008 (UTC)


 * Let's take the Mandelbrot set as an example. The Mandelbrot set is based on the family of maps $$f_c(z)=z^2+c$$ where c is a complex number. To determine whether the point representing a complex number c is in the Mandelbrot set you look at the behaviour of the critical point, 0, as you repeatedly apply the function $$f_c$$. This gives you the sequence of values
 * $$f_c(0)=c$$
 * $$f_c(c)=c^2+c$$
 * $$f_c(c^2+c)=(c^2+c)^2+c$$
 * etc.
 * For some values of c this sequence heads off towards infinity - these points are not in the Mandelbrot set. For other values of c this sequence wanders around without any pattern - these points are inside the Mandelbrot set. For some values of c the sequence eventually falls into a repeating pattern - for example, when c=i we have
 * $$f_c(0)=i$$
 * $$f_c(i)=(i)^2+i=-1+i$$
 * $$f_c(-1+i)=(-1+i)^2+i=-i$$
 * $$f_c(-i)=(-i)^2+i=-1+i$$
 * $$f_c(-1+i)=(-1+i)^2+i=-i$$
 * etc.
 * Here we reach a cycle with period 2, after the first 2 values. These "eventually periodic" points are called Misiurewicz points and they lie on the edge of the Mandelbrot set (although note that the "edge" of the Mandelbrot set has a very complicated structure). Gandalf61 (talk) 16:25, 13 March 2008 (UTC)
 * Ok, that makes more sense now, thanks.  D a n si m a n  ( talk | Contribs ) 00:19, 14 March 2008 (UTC)

types of Misiurewicz points
Hi, IMHO Misiurewicz point can not be an "interior point of the Mandelbrot set, with two external arguments ". I have reverted this edit. I can't ask the author because he has no user page. --Adam majewski (talk) 10:29, 2 November 2009 (UTC)

Hi, I concur (I wrote that text): Misiurewicz points cannot be interior points of the Mandelbrot set. That was a typo; I apologize. It was meant to be an "interior point of an arc within M", as currently written. The classification is simple: according to Douady-Hubbard theory (see the manuscripts by Douady/Hubbard, or Schleicher), every Misiurewicz point is the landing point of one or more external parameter rays. If it is one, then we have an endpoint; if it is three or more, then we have a branch point (and presumably the "spiral centers", whatever these are, are among them); and if there are exactly two rays landing, then this Misiurewicz point is not easily visible in the Mandelbrot set. But such points are there, and the are interior points of arcs within M. Examples are the landing points of the 5/12 and 7/12 rays. —Preceding unsigned comment added by DSS65 (talk • contribs) 22:36, 6 November 2009 (UTC)


 * Hi. I see from your user page in wiki that you are mathematician. Great. Experts are wellcome so thank you for your edits. (:-)).

I'm not an expert so I'm happy that you can improve this article.

Centers of the spirals are from this Mandelbrot page. So this seems to be an good source. I do not know any example of such point and I would like to put it here ( c value and number and angles of rays landing there).

"interior points of arcs within M". Could you expand it ( in article) and make an image.

If you could look and check also these pages :
 * external ray
 * wikibooks about mandelbrot set
 * wikibooks about Julia/Fatou set
 * my page about misiurewicz points.

Sorry for so many questions. Best regards. --Adam majewski (talk) 20:10, 7 November 2009 (UTC)


 * Hello, I added a few sentences about spirals. Actually, most Misiurewicz parameters are centers of spirals, even though this may not always be easily visible. Try any non-real Misiurewicz point and magnify at many levels, and you'll see things turning, perhaps slowly. -- I also added a sentence on 'interior points of arcs'; these are indeed difficult to see, but they are there nonetheless (such as the common endpoints of the rays at angles 5/12 and 7/12: I think you can make such pictures yourself).
 * As requested, I also had a quick look at the other sites you mentioned. External rays seem fine at first glance, and the others are more algorithms than mathematics, and that I leave to others. DSS65 (talk) 22:28, 7 November 2009 (UTC)


 * Thx. What means :

Regards. --Adam majewski (talk) 07:33, 8 November 2009 (UTC)
 * Tan Lei's aforementionen theorem
 * arc within the Mandelbrot set
 * principal Misiurewicz point of the limb
 * "all branch points of the Mandelbrot set are Misiurewicz points (plus, in a combinatorial sense, hyperbolic components represented by their centers)". Does it mean that hyperbolic components are Misiurewicz points in combinatorial sense ???

Preperiod and period
Hi. Preperiod and period are positive integers, but not all values are allowed. For example there is no Misiurewicz point for preperiod 1 and period 1. . What is a rule for choosing good values of preperiod and period ? --Adam majewski (talk) 09:03, 8 November 2009 (UTC)

The critical value can have any positive integer for period and for preperiod. Both have value 1 exactly for c=-2, where the critical values becomes fixed after one iteration. In general, if k is the period and m is the preperiod, then any angle r=a/(2^k)(2^m-1) has preperiod m and period k under multiplication by 2 modulo 1 (whenever a is an odd integer coprime to 2^m-1) and the corresponding parameter ray lands at a Misiurewicz parameter with preperiod m and period k (or possibly dividing k). The exact period of the cycle for such a Misiurewicz parameter can be discoverd by the kneading sequence of the angle (see e.g. the "Internal addresses" papers). DSS65 (talk) 23:50, 13 November 2009 (UTC)


 * Maybe I'm wrong but as I know c=-2 has preperiod 2 and period 1 ( it is also in article). Could you explain it ? --Adam majewski (talk) 08:36, 14 November 2009 (UTC)


 * Of course one has to specify whether one is talking about the preperiod of the critical _point_ or the critical _value_ (they obviously differ by 1, while their _periods_ are the same). For many reasons, things are more convenient for the critical _value_ (its properties transfer more directly to the Mandelbrot set), so this is why I was talking about the preperiodic of the critical value. If the critical value is periodic, then the critical point must be also periodic because (for degree 2 polynomials) it is the only preimage of the critical value. Therefore, the critical point can never have preperiod 1. That's the only exception (and it disappears if you discuss the preperiod of the critical value as proposed). Greetings, DSS65 (talk) 00:32, 19 November 2009 (UTC)


 * Thx for answer. This talk is very intresting. I have noted that difference, when I have checked iteration of Misiurewicz point under quadratic polynomial and iteration of external angle under period doubling map. ( If it is good comparison of course).

I think about adding values of preperiod and period to all examples, but I do not know all.

Could you also answer the questions above (types of Misiurewicz points) ? Your answer are great. --Adam majewski (talk) 19:07, 19 November 2009 (UTC)

Precision
Some Misiurewicz points are computed numerically and their values are aproximated. I think that word "near" should be replaced with $$value\pm error$$. --Adam majewski (talk) 08:30, 14 November 2009 (UTC)

Relation between Fegenbaum and Misiurerwicz points
Is it any relation ? TIA --Adam majewski (talk) 09:54, 13 November 2016 (UTC)