In mathematics, a Misiurewicz point is a parameter in the Mandelbrot set for which the critical point is strictly preperiodic. By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly preperiodic.
Mathematical notation
A parameter is a Misiurewicz point if it satisfies the equations and so : where :
Misiurewicz points are named after the Polish-American mathematicianMichał Misiurewicz. Note that the term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent, 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 this term is used in complex dynamics; a more appropriate term would be Misiurewicz–Thurston points.
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 which has a single critical point at. The Misiurewicz points of this family of maps are roots of the equations , where :
For example, the Misiurewicz points with k=2 and n=1, denoted by M2,1, 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 M2,1 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 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 is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling. The Mandelbrot set and Julia set 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
non-branch points with exactly 2 external arguments : 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, all branch points of the Mandelbrot set are Misiurewicz points. Many 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.
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 :
is a tip of the filament
Its critical orbits is
landing point of the external ray for the angle = 1/6
Point
is the end-point of main antenna of Mandelbrot set
