A Misiurewicz point is a point $c$ in the parameter space of a family of discrete dynamical systems $z\_\{n+1\}\; =\; f\_c(z\_n)\; ,$ (indexed by a real or complex parameter) such that a critical point of $f\_c$ is pre-periodic i.e. there are positive integers k and n such that:

$f\_c^\{(k-1)\}(z\_0)\; neq\; f\_c^\{(k+n-1)\}(z\_0)\; ,$

$f\_c^\{(k)\}(z\_0)\; =\; f\_c^\{(k+n)\}(z\_0)\; ,$

where $z\_0\; ,$ is a critical point of $f\_c$.

Misiurewicz points are named after mathematician Michał Misiurewicz

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 $P\_c(z)=z^2+c\; ,$ which has a single critical point at $z\; =\; 0,$. The Misiurewicz points of this family of maps are the roots of the equations

$P\_c^\{(k)\}(0)\; =\; P\_c^\{(k+n)\}(0)$,

where :

• k is the pre-period
• n is the period
• $P\_c^\{(n)\}\; =\; P\_c\; (P\_c^\{(n-1)\}),$ denotes the n-fold composition of $P\_c(z)=z^2+c,$ with itself i.e. the n^th iteration of $P\_c,$.

For example, the Misiurewicz points with k=2 and n=1, denoted by M_2,1, are roots of

$P\_c^\{(2)\}(0)\; =\; P\_c^\{(3)\}(0)$

$Rightarrow\; c^2+c=(c^2+c)^2+c$

$Rightarrow\; c^4+2c^3=0$.

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 are boundary points = All Misiurewicz points belong to the boundary of the Mandelbrot set. Misiurewicz points are dense at the boundary of Mandelbrot set

If $c,$ is a Misiurewicz point, then the associated filled Julia set is equal to Julia set, it means filled Julia set has no interior.

If $c,$ is a Misiurewicz point, then the corresponding periodic cycle is repelling.

Mandelbrot set and Julia set $J\_c,$ are locally similar around Misiurewicz points. Mandelbrot set is self-similar around Misiurewicz points

Misiurewicz points can be :

• branch points (= points where filaments meet.) with 3 or more external arguments (angles )
• end points of :
  • the filaments
  • main antenna
  with 1 external argument
• centers of spirals

External arguments of Misiurewicz points are rational numbers with even denominator

Examples of Misiurewicz points of complex quadratic mapping

Point $c\; =\; -2,$, which is the end-point of main antenna of Mandelbrot set.

This diagram shows iteration of critical point of complex quadratic polynomial (it means $z\; =\; 0,$ )
for $c\; =\; -2,$.
Notice that it is z-plane ( dynamical plane) not c-plane ( parameter plane) and point $z\; =\; -2,$ is not the same point as $c\; =\; -2,$.
One can see that critical point $z\; =\; 0,$ is preperiodic with preperiod 2 and period 1.
Orbit of critical point = $\{\; 0\; ,\; -2,\; 2,\; 2,\; 2,\; ...\; \}\; ,$
Symbolic sequence = C L R R R ...

Point $c\; =\; -2\; =M\_\{2,1\},$ is landing point of only one external ray (parameter ray) of angle 1/2 .

Point $c\; =\; -0.1011\; +0.9563*i\; =M\_\{4,1\},$ is a principial Misiurewicz point of the 1/3 limb. It has 3 external rays 9/56, 11/56 and 15/56.

Point $c=\; -0.77568377+0.13646737*i\; ,$ is a Misiurewicz point which is a center of a two-arms spiral.

Computing Misiurewicz points of complex quadratic mapping in Maxima

Define polynomial.

P(n):=if n=0 then 0 else P(n-1)^2+c;

Define a function whose roots are Misiurewicz points, and find them.

M(preperiod,period):=allroots(%i*P(preperiod+period)-%i*P(preperiod));

Examples of use :

(%i6) M(2,1);

(%o6) [c=-2.0,c=0.0]

(%i7) M(2,2);

(%o7) [c=-1.0*%i,c=%i,c=-2.0,c=-1.0,c=0.0]

References

Further reading

• Michał Misiurewicz (1981), "Absolutely continuous measures for certain maps of an interval" Publications Mathématiques de l'IHÉS, 53 (1981), p. 17-51
• G. Pastor, M. Romera and F. Montoya (1996). "On the calculation of Misiurewicz patterns in one-dimensional quadratic maps" Physica A, 232 (1996), 536-553
• G. Pastor, M. Romera, G. Álvarez and F. Montoya (2001). "Misiurewicz point patterns generation in one-dimensional quadratic maps", Physica A, 292 (2001), 207-230
• G. Pastor, M. Romera, G. Álvarez and F. Montoya (2003). "How to work with one-dimensional quadratic maps", Chaos, Solitons and Fractals, 18 (2003), 899-915
• M. Romera, G. Pastor and F. Montoya (1996), Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535.

External links

• Preperiodic (Misiurewicz) points in the Mandelbrot set by Evgeny Demidov
• M & J-sets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel
• The boundary of the Mandelbrot set by Michael Frame, Benoit Mandelbrot, and Nial Neger
• Misiurewicz Point of the logistic map by J. C. Sprott