Definitions

mispoint

Misiurewicz point

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 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^{(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 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 :

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

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