Talk:Misiurewicz point
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[edit] Rewritten
I have rewrite the article. --Adam majewski 10:59, 17 June 2007 (UTC)
[edit] definition
Hi what is the meaning and source of this statement :
?
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
- 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
- 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
, 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 
?
4. ( new question) How to compute angles of external rays landing on ?
Adam majewski 14:57, 23 July 2007 (UTC)
[edit] 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, Dansiman (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 fc(z) = z2 + 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 fc. This gives you the sequence of values
- fc(0) = c
- fc(c) = c2 + c
- fc(c2 + c) = (c2 + 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
- fc(0) = i
- fc(i) = (i)2 + i = − 1 + i
- fc( − 1 + i) = ( − 1 + i)2 + i = − i
- fc( − i) = ( − i)2 + i = − 1 + i
- fc( − 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)
