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 ${\displaystyle f_{c}}$.

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 neighborhood 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).

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.

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[10]
• Landing point of the external ray for the angle = 1/6
• Its critical orbits is ${\displaystyle \{0,i,i-1,-i,i-1,-i...\}}$ [11]

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

• Is the endpoint of main antenna of Mandelbrot set[12]
• Is landing point of only one external ray (parameter ray) of angle 1/2
• Its critical orbit:
  • Is ${\displaystyle \{0,-2,2,2,2,...\}}$ [11]
  • Symbolic sequence = C L R R R ...
  • Pre-period is 2 and period is 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 a principal Misiurewicz point of the 1/3 limb
• It 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 ${\displaystyle M_{23,2}}$. 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 pre-period ${\displaystyle k=23}$ 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 pre-period ${\displaystyle k=3}$ and period ${\displaystyle n=1}$

• Arithmetic dynamics
• Feigenbaum point
• Dendrite (mathematics)

Wikibooks has a book on the topic of: Fractals

Wikimedia Commons has media related to 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