Misiurewicz point

In mathematics, a Misiurewicz point is a parameter in the Mandelbrot set (the parameter space of quadratic polynomials) for which the critical point 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 formula_1 is a Misiurewicz point formula_2 if it satisfies the equations
and 
so :
where :
Name.
Misiurewicz points are named after the Polish-American mathematician Michał Misiurewicz.
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. 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 conjugation any quadratic polynomial can be transformed into a map of the form formula_13 which has a single critical point at formula_14. The Misiurewicz points of this family of maps are roots of the equations
(subject to the condition that the critical point is not periodic), 
where :
For example, the Misiurewicz points with "k"=2 and "n"=1, denoted by "M"2,1, are roots of
The root "c"=0 is not a Misiurewicz point because the critical point is a fixed point when "c"=0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point "M"2,1 at "c" = −2.
Properties of Misiurewicz points of complex quadratic mapping.
Misiurewicz points belong to the boundary of the Mandelbrot set. Misiurewicz points are dense in the boundary of the Mandelbrot set.
If formula_1 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.
If formula_1 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).
The Mandelbrot set and Julia set formula_24 are locally asymptotically self-similar around Misiurewicz points.
Types.
Misiurewicz points can be classified according to number of external rays that land on them :, points where branches meet
According to the Branch Theorem of the Mandelbrot set, all branch points of the Mandelbrot set are Misiurewicz points (plus, in a combinatorial sense, hyperbolic components represented by their centers).
Many (actually, most) Misiurewicz parameters in the Mandelbrot set look like `centers of spirals'. 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 magnification.)
External arguments.
External arguments of Misiurewicz points, measured in turns are :
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.
Point formula_25 :
Point formula_27 
Notice that it is z-plane (dynamical plane) not c-plane (parameter plane) and point formula_29 is not the same point as formula_30.
Point formula_31 is landing point of only one external ray ( parameter ray) of angle 1/2 .
Non-branch points.
Point formula_32 is near a Misiurewicz point formula_33. It is 
Point formula_39 is near a Misiurewicz point formula_40, 
Branch points.
Point formula_45 