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Misiurewicz point

A Misiurewicz point is a point $c$ in the parameter space of a family of discrete dynamical systems $z_\left\{n+1\right\} = f_c\left(z_n\right) ,$ (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:

$f_c^\left\{\left(k-1\right)\right\}\left(z_0\right) neq f_c^\left\{\left(k+n-1\right)\right\}\left(z_0\right) ,$
$f_c^\left\{\left(k\right)\right\}\left(z_0\right) = f_c^\left\{\left(k+n\right)\right\}\left(z_0\right) ,$

where $z_0 ,$ is a critical point of $f_c$.

Misiurewicz points are named after mathematician Michał Misiurewicz

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

$P_c^\left\{\left(k\right)\right\}\left(0\right) = P_c^\left\{\left(k+n\right)\right\}\left(0\right)$,

where :

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

For example, the Misiurewicz points with k=2 and n=1, denoted by M2,1, are roots of

$P_c^\left\{\left(2\right)\right\}\left(0\right) = P_c^\left\{\left(3\right)\right\}\left(0\right)$
$Rightarrow c^2+c=\left(c^2+c\right)^2+c$
$Rightarrow c^4+2c^3=0$.

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 M2,1 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 : 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 = $\left\{ 0 , -2, 2, 2, 2, ... \right\} ,$
Symbolic sequence = C L R R R ...

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

Point $c = -0.1011 +0.9563*i =M_\left\{4,1\right\},$ 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]`