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]

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.

Properties of Misiurewicz points of complex quadratic mapping

Misiurewicz points belong to, and are dense in, the boundary of the Mandelbrot set.^[4][5]

If ${\displaystyle c}$ is a Misiurewicz point, then the associated filled Julia set is equal to the Julia set and means the filled Julia set has no interior.

If ${\displaystyle c}$ is a Misiurewicz point, then in the corresponding Julia set all periodic cycles are repelling (in particular the cycle that the critical orbit falls onto).

The Mandelbrot set and Julia set ${\displaystyle J_{c))$ are locally asymptotically self-similar around Misiurewicz points.^[6]

Types

Misiurewicz points in the context of the Mandelbrot set can be classified based on several criteria. One such criterion is the number of external rays that converge on such a point.^[4] Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more external arguments (or angles). Non-branch points have exactly two external rays (these correspond to points lying on arcs within the Mandelbrot set). These non-branch points are generally more subtle and challenging to identify in visual representations. End points, or branch tips, have only one external ray converging on them. Another criterion for classifying Misiurewicz points is their appearance within a plot of a subset of the Mandelbrot set. Misiurewicz points can be found at the centers of spirals as well as at points where two or more branches meet.^[7] According to the Branch Theorem of the Mandelbrot set,^[5] all branch points of the Mandelbrot set are Misiurewicz points.^[4][5]

Most Misiurewicz parameters within the Mandelbrot set exhibit a "center of a spiral".^[8] This occurs due to the behavior at a Misiurewicz parameter where the critical value jumps onto a repelling periodic cycle after a finite number of iterations. At each point during the cycle, the Julia set exhibits asymptotic self-similarity through complex multiplication by the derivative of this cycle. If the derivative is non-real, it implies that the Julia set near the periodic cycle has a spiral structure. Consequently, a similar spiral structure occurs in the Julia set near the critical value, and by Tan Lei's theorem, also in the Mandelbrot set near any Misiurewicz parameter for which the repelling orbit has a non-real multiplier. The visibility of the spiral shape depends on the value of this multiplier. The number of arms in the spiral corresponds to the number of branches at the Misiurewicz parameter, which in turn equals the number of branches at the critical value in the Julia set. Even the principal Misiurewicz point in the 1/3-limb, located at the end of the parameter rays at angles 9/56, 11/56, and 15/56, is asymptotically a spiral with infinitely many turns, although this is difficult to discern without magnification.^{[citation needed]}

External arguments

External arguments of Misiurewicz points, measured in turns are:

• Rational numbers
• Proper fractions with an even denominator
  • Dyadic fractions with denominator ${\displaystyle =2^{b))$ and finite (terminating) expansion:
    $${\displaystyle {\frac {1}{2))_{10}=0.5_{10}=0.1_{2))$$

    
  • Fractions with a denominator ${\displaystyle =a\cdot 2^{b))$ and repeating expansion:^[9]
    $${\displaystyle {\frac {1}{6))_{10}={\frac {1}{2\times 3))_{10}=0.16666..._{10}=0.0(01)..._{2}.}$$

    

where: a and b are positive integers and b is odd, subscript number shows base of numeral system.

Examples of Misiurewicz points of complex quadratic mapping

End points

Point ${\displaystyle c=M_{2,2}=i}$ is considered an end point as it is a tip of a filament,^[10] and the landing point of the external ray for the angle 1/6. Its critical orbit is ${\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 the main antenna of the Mandelbrot set.^[12] and the landing point of only one external ray (parameter ray) of angle 1/2. It is also considered 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

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.

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 ${\displaystyle n=1}$.