Misiurewicz point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval[1] for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself).
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.
These points are named after the Polish-American mathematician Michał Misiurewicz, who was the first to study them.
This meaning is firmly established in the context of the dynamics of iterated interval maps.
[3] Only in very special cases does a quadratic polynomial have a strictly periodic and unique critical point.
In this restricted sense, the term is used in complex dynamics; a more appropriate one would be Misiurewicz–Thurston points (after William Thurston, who investigated post-critically finite rational 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
Misiurewicz points belong to, and are dense in, the boundary of the Mandelbrot set.
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).
are locally asymptotically self-similar around Misiurewicz points.
[6] Misiurewicz points in the context of the Mandelbrot set can be classified based on several criteria.
One such criterion is the number of external rays that converge on such a point.
[4] Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more external arguments (or angles).
These non-branch points are generally more subtle and challenging to identify in visual representations.
End points, or branch tips, have only one external ray converging on them.
Another criterion for classifying Misiurewicz points is their appearance within a plot of a subset of the Mandelbrot set.
[4][5] Most Misiurewicz parameters within the Mandelbrot set exhibit a "center of a spiral".
[8] This occurs due to the behavior at a Misiurewicz parameter where the critical value jumps onto a repelling periodic cycle after a finite number of iterations.
If the derivative is non-real, it implies that the Julia set near the periodic cycle has a spiral structure.
Consequently, a similar spiral structure occurs in the Julia set near the critical value, and by Tan Lei's theorem, also in the Mandelbrot set near any Misiurewicz parameter for which the repelling orbit has a non-real multiplier.
The visibility of the spiral shape depends on the value of this multiplier.
in the 1/3-limb, located at the end of the parameter rays at angles 9/56, 11/56, and 15/56, is asymptotically a spiral with infinitely many turns, although this is difficult to discern without magnification.
[citation needed] External arguments of Misiurewicz points, measured in turns are: The subscript number in each of these expressions is the base of the numeral system being used.
is considered an end point as it is the endpoint of the main antenna of the Mandelbrot set.
It is also considered an end point because its critical orbit is
This can be seen because it is a center of a two-arms spiral, the landing point of 2 external rays with angles:
