A preperiodic orbit.

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). By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. 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.^[2]

Principal Misiurewicz point of the wake 1/31

Mathematical notation

A parameter ${\displaystyle c}$ is a Misiurewicz point ${\displaystyle M_{k,n}}$ if it satisfies the equations:

${\displaystyle f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})}$

and:

${\displaystyle f_{c}^{(k-1)}(z_{cr})\neq f_{c}^{(k+n-1)}(z_{cr})}$

so:

${\displaystyle M_{k,n}=c:f_{c}^{(k)}(z_{cr})=f_{c}^{(k+n)}(z_{cr})}$

where:

• ${\displaystyle z_{cr}}$ is a critical point of ${\displaystyle f_{c}}$,
• ${\displaystyle k}$ and ${\displaystyle n}$ are positive integers,
• ${\displaystyle f_{c}^{(k)}}$ denotes the ${\displaystyle k}$-th iterate of ${\displaystyle f_{c}}$.

Name

The term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent; that is, in which there exists a neighbourhood for every critical point that is not visited by the orbit of this critical point. 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).

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 ${\displaystyle P_{c}(z)=z^{2}+c}$ which has a single critical point at ${\displaystyle z=0}$. The Misiurewicz points of this family of maps are roots of the equations:

${\displaystyle P_{c}^{(k)}(0)=P_{c}^{(k+n)}(0),}$

Subject to the condition that the critical point is not periodic, where:

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

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

${\displaystyle {\begin{aligned}&P_{c}^{(2)}(0)=P_{c}^{(3)}(0)\\\Rightarrow {}&c^{2}+c=(c^{2}+c)^{2}+c\\\Rightarrow {}&c^{4}+2c^{3}=0.\end{aligned}}}$

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.

Examples of Misiurewicz points of complex quadratic mapping

End points

Orbit of critical point ${\displaystyle z=0}$ under ${\displaystyle f_{-2}}$

${\displaystyle c=M_{2,1}}$

Point ${\displaystyle c=M_{2,2}=i}$ is considered an end point as it is a tip of the filament^[10], the landing point of the external ray for the angle =1/6, and its critical orbits are: ${\displaystyle \{0,i,i-1,-i,i-1,-i...\}}$ ^[11]

Point ${\displaystyle c=M_{2,1}=-2}$ is considered an end point as it is the endpoint of main antenna of Mandelbrot set^[12] and the landing point of only one external ray (parameter ray) of angle 1/2. It is also considedered an end point because its critical orbit is ${\displaystyle \{0,-2,2,2,2,...\}}$ ^[11], following the Symbolic sequence =C L R R R ... with a pre-period of 2 and period of 1

Branch points

Zoom around principal Misiurewicz point for periods from 2 to 1024

${\displaystyle c=M_{4,1}}$

Point ${\displaystyle c=-0.10109636384562...+i\,0.95628651080914...=M_{4,1}}$ is considered a branch point because it is a principal Misiurewicz point of the 1/3 limb and has 3 external rays: 9/56, 11/56 and 15/56.

Other points

These are points which are not-branch and not-end points.

${\displaystyle c=M_{23,2}}$

Point ${\displaystyle c=-0.77568377+i\,0.13646737}$ is near a Misiurewicz point ${\displaystyle M_{23,2}}$. This can be seen because it is a center of a two-arms spiral, the landing point of 2 external rays with angles: ${\displaystyle {\frac {8388611}{25165824}}}$ and ${\displaystyle {\frac {8388613}{25165824}}}$ where the denominator is ${\displaystyle 3*2^{23}}$, and has a preperiodic point with pre-period ${\displaystyle k=23}$ and period ${\displaystyle n=2}$

Point ${\displaystyle c=-1.54368901269109}$ is near a Misiurewicz point ${\displaystyle M_{3,1}}$, as it is the landing point for pair of rays: ${\displaystyle {\frac {5}{12}}}$, ${\displaystyle {\frac {7}{12}}}$ and has pre-period ${\displaystyle k=3}$ and period ${\displaystyle n=1}$.

See also

• Arithmetic dynamics
• Feigenbaum point
• Dendrite (mathematics)

References

Further reading

External links

Wikibooks has a book on the topic of: Fractals

• Preperiodic (Misiurewicz) points in the Mandelbrot set by Evgeny Demidov
• M & J-sets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel
• Misiurewicz Point of the logistic map by J. C. Sprott