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In mathematics, a '''Misiurewicz point''' is a parameter in the [[Mandelbrot set]] (the [[parameter space]] of quadratic polynomials) for which the [[critical point (mathematics)|critical point]]<!-- Term is undefined in this context --> is strictly preperiodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term ''Misiurewicz point'' is also used for parameters in a [[Multibrot set]] where the unique critical point is strictly preperiodic. (This term makes less sense for maps in greater generality that have more than one (free) critical point because some critical points might be periodic and others not.)

==Mathematical notation ==
A parameter <math>c</math> is a Misiurewicz point <math>M_{k,n} \,</math> if it satisfies the equations
:<math>f_c^{(k)}(z_{cr}) = f_c^{(k+n)}(z_{cr}) \,</math>

and 
:<math> f_c^{(k-1)}(z_{cr}) \neq f_c^{(k+n-1)}(z_{cr})\,</math>

so :
:<math>M_{k,n} = c : f_c^{(k)}(z_{cr}) = f_c^{(k+n)}(z_{cr}) \,</math>

where :
*<math>z_{cr} \,</math> is a [[critical point (mathematics)|critical point]] of <math>f_c\,</math>,
*<math>k\,</math> and <math>n\,</math> are positive integers,
and <math> f_c^{k} </math> denotes the <math> k</math>-th iterate of <math> f_c</math>.

==Name==
Misiurewicz points are named  after the Polish-American [[mathematician]] [[Michał Misiurewicz]].<ref>[http://www.math.iupui.edu/~mmisiure/ Michał Misiurewicz home page], Indiana University-Purdue University Indianapolis</ref>

Note that the term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent (that is, there is a neighborhood of every critical point that is not visited by the orbit of this critical point), and this meaning is firmly established in the context of dynamics of iterated interval maps.<ref>Wellington de Melo, Sebastian van Strien, "One-dimensional dynamics". Monograph, Springer Verlag (1991)</ref> The case that for a quadratic polynomial the unique critical point is strictly preperiodic is only a very special case; in this restricted sense (as described above) this term is used in complex dynamics; a more appropriate term would be '''Misiurewicz-Thurston points''' (after [[William Thurston]] who investigated postcritically finite rational maps).

==Quadratic maps==
A [[complex quadratic polynomial]] has only  one critical point. By a suitable [[conjugacy class|conjugation]] any quadratic polynomial can be transformed into a map of the form <math> P_c(z)=z^2+c \,</math> which has a single critical point at <math> z = 0\,</math>. The Misiurewicz points of this family of maps are [[root of a function|roots]] of the equations

:<math>P_c^{(k)}(0) = P_c^{(k+n)}(0)</math>,

(subject to the condition that the critical point is not periodic), 
where :
*''k'' is the pre-period
*''n'' is the period
*<math>P_c^{(n)} = P_c ( P_c^{(n-1)})\,</math> denotes  the ''n''-fold [[Function composition|composition]] of <math>P_c(z)=z^2+c\,</math> with itself i.e. the ''n''<sup>th</sup> [[Iterated function|iteration]] of <math>P_c\,</math>.

For example, the Misiurewicz points with ''k''=2 and ''n''=1, denoted by ''M''<sub>2,1</sub>, are roots of

:<math>P_c^{(2)}(0) = P_c^{(3)}(0)</math>
:<math>\Rightarrow c^2+c=(c^2+c)^2+c</math>
:<math>\Rightarrow c^4+2c^3=0</math>.

The root ''c''=0 is not a Misiurewicz point because the critical point is a [[fixed point (mathematics)|fixed point]] when ''c''=0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point ''M''<sub>2,1</sub> at ''c'' = &minus;2.

===Properties of Misiurewicz points of  complex quadratic mapping===
Misiurewicz points belong to the [[Boundary (topology)|boundary]] of the [[Mandelbrot set]]. Misiurewicz points are [[Dense set|dense]] in the [[Boundary (topology)|boundary]] of the [[Mandelbrot set]].<ref name=Douady>Adrien Douady, John Hubbard, "Etude dynamique des polynômes complexes", prépublications mathématiques d'Orsay, 1982/1984</ref><ref name=Schleicher>Dierk Schleicher, "On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets", in: M. Lapidus, M. van Frankenhuysen (eds): Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proceedings of Symposia in Pure Mathematics 72, American Mathematical Society (2004), 477–507 or [http://arxiv.org/abs/math/9902155 online paper from arXiv.org]</ref>

If <math>c\,</math> is a Misiurewicz point, then the associated [[filled Julia set]] is equal to the [[Julia set]], and means the [[filled Julia set]] has no [[Interior (topology)|interior]].

If <math>c\,</math> is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling (in particular the cycle that the critical orbit falls onto).

[[Mandelbrot set]] and [[Julia set]] <math>J_c\,</math> are locally asymptotically [[Similarity (geometry)|similar]] around Misiurewicz points. [[Mandelbrot set]] is [[Self-similarity|self-similar]] around Misiurewicz points <ref>[http://projecteuclid.org/euclid.cmp/1104201823 Lei.pdf]  Tan Lei, "Similarity between the Mandelbrot set and Julia Sets", Communications in Mathematical Physics 134 (1990), pp. 587-617.</ref>

====Types====
Misiurewicz points can be classified according to number of external rays that land on them :<ref name=Douady/>
* branch points  ( =  points that disconnect the Mandelbrot set into at least three components.) with 3 or more [[external ray|external arguments ( angles )]]
* non-branch points  with exactly 2 external arguments ( = interior points of arcs within the Mandelbrot set) : these points are less conspicuous and thus not so easily to find on pictures.
* end points with  1 external argument

According to the Branch Theorem of the Mandelbrot set,<ref name=Schleicher/> all branch points of the Mandelbrot set are Misiurewicz points (plus, in a combinatorial sense, hyperbolic components represented by their centers).<ref name=Douady/><ref name=Schleicher/>

Many (actually, most) Misiurewicz parameters in the Mandelbrot set look like `centers of spirals'.<ref>[http://classes.yale.edu/fractals/MandelSet/MandelBoundary/Mis.html The boundary of the Mandelbrot set] by Michael Frame, Benoit Mandelbrot, and Nial Neger</ref> The explanation for this is the following: at a Misiurewicz parameter, the critical value jumps onto a repelling periodic cycle after finitely many iterations; at each point on the cycle, the Julia set is asymptotically self-similar by a complex multiplication by the derivative of this cycle. If the derivative is non-real, then this implies that the Julia set, near the periodic cycle, has a spiral structure. A similar spiral structure thus occurs in the Julia set near the critical value and, by Tan Lei's aforementioned theorem, also in the Mandelbrot set near any Misiurewicz parameter for which the repelling orbit has non-real multiplier. Depending on the value of the multiplier, the spiral shape can seem more or less pronounced. The number of the arms at the spiral equals the number of branches at the Misiurewicz parameter, and this equals the number of branches at the critical value in the Julia set. (Even the `principal Misiurewicz point in the 1/3-limb', at the end of the parameter rays at angles 9/56, 11/56, and 15/56, turns out to be asymptotically a spiral, with infinitely many turns, even though this is hard to see without maginification.)

====External arguments====
[[External ray|External arguments]] of Misiurewicz points,  measured in [[Turn (geometry)|turns]] are :
{{unordered list
|1= [[rational numbers]]
|2= [[Fraction_(mathematics)#Vulgar.2C_proper.2C_and_improper_fractions|proper fraction]] with [[Even and odd numbers|even]] [[Vulgar fraction|denominator]]
*[[dyadic fraction]]s with denominator <math> = 2^b\,</math> and finite ( [[terminating decimal|terminating]] ) expansion, like :
: <math>\frac{1}{2}_{10} = 0.5_{10} =   0.1_2 \,</math>
*fraction with denominator <math> = a*2^b\,</math> and [[repeating decimal|repeating expansion]] like :
:<math>\frac{1}{6}_{10} = \frac{1}{2*3}_{10}= 0,16666..._{10} = 0.0(01)..._2\,</math>.<ref>[http://math.unipa.it/~grim/21_project/CiechYeung.pdf BINARY DECIMAL NUMBERS AND DECIMAL NUMBERS OTHER THAN BASE TEN by Thomas Kim-wai YEUNG and Eric Kin-keung POON]</ref>
}}
where: a and b are positive integers and b is odd,  subscript number shows base of [[numeral system]].

===Examples of Misiurewicz points of  complex quadratic mapping===

====End points====
[[Image:Prep21.jpg|right|thumb| Orbit of critical point <math>z = 0\,</math> under <math> f_{-2}\,</math>]]
[[Image:MIS1.jpg|thumb|right|<math> c =M_{2,1}\,</math>]]
Point <math>c =  M_{2,2} = i \,</math>  :
* is a tip of the filament<ref>[http://www.mrob.com/pub/muency/terminalpoint.html Tip of the filaments by Robert P. Munafo]</ref>
* Its critical orbits is  <math> \{0, i, i-1, -i, i-1, -i...\} \,</math> <ref name="ibiblio.org">[http://www.ibiblio.org/e-notes/MSet/CPeriod.htm Preperiodic (Misiurewicz) points in the Mandelbrot se] by Evgeny Demidov</ref>	
	
Point <math> c = M_{2,1} = -2\,</math>  
* is the end-point of main antenna of [[Mandelbrot set]] <ref>[http://www.mrob.com/pub/muency/tip.html tip of main antennae by Robert P. Munafo]</ref>
* Its critical orbits is  <math> \{ 0 , -2, 2, 2, 2, ... \} \,</math> <ref name="ibiblio.org"/>	
* [[Symbolic dynamics|Symbolic sequence]] = C L R R R ...
* preperiod is 2 and period 1

Notice that it is z-plane ([[phase space|dynamical plane]]) not c-plane ([[parameter space|parameter plane]]) and point <math> z = -2\,</math> is not the same point as <math> c = -2\,</math>.

Point <math> c = -2 =M_{2,1}\,</math> is  landing point of only one [[external ray]] ( parameter ray) of angle 1/2 .

====Non-Branch points====

[[File:SpiralM23 2.png|thumb|right|<math> c =M_{23,2}\,</math>]]

Point <math>c= -0.77568377+0.13646737*i \,</math> is near a Misiurewicz point <math>M_{23,2}\,</math>. It is 
* a center of a two-arms spiral
* a landing point of 2 external rays with angles : <math>\frac{8388611}{25165824}\,</math>  and  <math>\frac{8388613}{25165824}\,</math> where denominator is <math>3*2^{23}\,</math>
* preperiodic point with preperiod <math>k = 23\,</math> and period <math>n = 2\,</math>

Point <math>c = -1.54368901269109 \,</math> is near a Misiurewicz point <math>M_{3,1}\,</math>, 
* which is landing point for pair of rays : <math>\frac{5}{12}\,</math>, <math> \frac{7}{12}\,</math>
* has preperiod <math>k = 3\,</math> and period <math>n = 1\,</math>

====Branch points====

:[[Image:Misi13limb.jpg|thumb|right|<math> c =M_{4,1}\,</math>]]

Point <math> c = -0.1010... +0.9562...*i =M_{4,1}\,</math> is a principal Misiurewicz point of the 1/3 limb. It has 3 [[external ray]]s  9/56, 11/56 and 15/56.

== See also==
*[[Arithmetic dynamics]]

==References==
{{reflist}}
{{refbegin}}
*Michał Misiurewicz (1981), [http://www.numdam.org/item?id=PMIHES_1981__53__17_0 "Absolutely continuous measures for certain maps of an interval"]. Publications Mathématiques de l'IHÉS, 53 (1981), p.&nbsp;17-51
{{refend}}

==External links==
{{Wikibooks|Fractals }}
*[http://www.ibiblio.org/e-notes/MSet/CPeriod.htm Preperiodic (Misiurewicz) points in the Mandelbrot set] by Evgeny Demidov
*[http://www.ibiblio.org/e-notes/MSet/Lei90.htm M & J-sets similarity for preperiodic points. Lei's theorem ] by [[Douglas C. Ravenel]]
*[http://sprott.physics.wisc.edu/chaos/mispoint.htm Misiurewicz Point] of the [[logistic map]] by [http://sprott.physics.wisc.edu/sprott.htm J. C. Sprott]

{{DEFAULTSORT:Misiurewicz Point}}
[[Category:Fractals]]
[[Category:Systems theory]]
[[Category:Dynamical systems]]