# Misiurewicz point

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 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 nth iteration of ${\displaystyle P_{c}}$.

For example, the Misiurewicz points with k= 2 and n= 1, denoted by M2,1, 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 M2,1 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 can be classified according to several criteria. One such criterion is the number of external rays that land 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 arguments (points of arcs within the Mandelbrot set). These points are less conspicuous and not easily found in pictures.
• Branch tips, or end points, will have one external argument.

Another criterion is how such a point looks within a plot of part of the 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 in the Mandelbrot set look like "centers of spirals".[8] The explanation for this is that at a Misiurewicz parameter, the critical value jumps onto a repelling periodic cycle after finitely many iterations. At each point during the cycle, the Julia set is asymptotically self-similar by a complex multiplication by the derivative of this cycle. If the derivative is non-real, this implies that the Julia set near the periodic cycle has a spiral structure. Thus, 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. Depending on the value of this multiplier, the spiral shape can seem either more or less pronounced. The number of arms at the spiral equals the number of branches at the Misiurewicz parameter, and this equals the number of branches at the critical value in the Julia set. Even the principal Misiurewicz point in the 1/3-limb, at the end of the parameter rays at angles 9/56, 11/56, and 15/56, turns out to be asymptotically a spiral with infinitely many turns, even though this is hard to see without magnification.

#### 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 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

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.

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 ${\displaystyle n=1}$

## References

1. ^ Diaz-Ruelas, A.; Baldovin, F.; Robledo, A. (19 January 2022). "Logistic map trajectory distributions:Renormalization-group, entropy, and criticality at the transition to chaos". Chaos: An Interdisciplinary Journal of Nonlinear Science. Chaos 31, 033112 (2021). 31 (3): 033112. doi:10.1063/5.0040544. PMID 33810710. S2CID 231933949.