Misiurewicz point

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 ${\displaystyle f_{c}}$.

Misiurewicz points are named after the Polish-American mathematician Michał Misiurewicz.[1]

Note that the term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent (that is, there is a neighborhood of every critical point that is not visited by the orbit of this critical point), and this meaning is firmly established in the context of dynamics of iterated interval maps.[2] The case that for a quadratic polynomial the unique critical point is strictly preperiodic is only a very special case; in this restricted sense (as described above) this term is used in complex dynamics; a more appropriate term would be Misiurewicz–Thurston points (after William Thurston, who investigated postcritically finite rational maps).

• hub ( in case of branch point)

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

${\displaystyle P_{c}^{(2)}(0)=P_{c}^{(3)}(0)}$

${\displaystyle \Rightarrow c^{2}+c=(c^{2}+c)^{2}+c}$

${\displaystyle \Rightarrow c^{4}+2c^{3}=0}$.

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.

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 a tip of the filament[8]
• Its critical orbits is ${\displaystyle \{0,i,i-1,-i,i-1,-i...\}}$ [9]
• landing point of the external ray for the angle = 1/6

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

• is the end-point of main antenna of Mandelbrot set [10]
• Its critical orbits is ${\displaystyle \{0,-2,2,2,2,...\}}$ [9]
• Symbolic sequence = C L R R R ...
• preperiod is 2 and period 1

Notice that it is z-plane (dynamical plane) not c-plane (parameter plane) and point ${\displaystyle z=-2}$ is not the same point as ${\displaystyle c=-2}$.

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

Non-branch points

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

Point ${\displaystyle c=-0.77568377+0.13646737*i}$ is near a Misiurewicz point ${\displaystyle M_{23,2}}$. It is

• a center of a two-arms spiral
• a landing point of 2 external rays with angles : ${\displaystyle {\frac {8388611}{25165824}}}$ and ${\displaystyle {\frac {8388613}{25165824}}}$ where denominator is ${\displaystyle 3*2^{23}}$
• preperiodic point with preperiod ${\displaystyle k=23}$ and period ${\displaystyle n=2}$

Point ${\displaystyle c=-1.54368901269109}$ is near a Misiurewicz point ${\displaystyle M_{3,1}}$,

• which is landing point for pair of rays : ${\displaystyle {\frac {5}{12}}}$, ${\displaystyle {\frac {7}{12}}}$
• has preperiod ${\displaystyle k=3}$ and period ${\displaystyle n=1}$

Branch points

Zoom around principal Misiurewicz point for periods from 2 to 1024

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

Point ${\displaystyle c=-0.1010...+0.9562...*i=M_{4,1}}$

• is a principal Misiurewicz point of the 1/3 limb
• it has 3 external rays: 9/56, 11/56 and 15/56.

• Arithmetic dynamics
• Feigenbaum point
• Dendrite (mathematics)

Wikibooks has a book on the topic of: Fractals

Wikimedia Commons has media related to Category:Misiurewicz point.

• 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