Misiurewicz point

In mathematics, a Misiurewicz point is a parameter in the Mandelbrot set (the parameter space of quadratic polynomials) for which the critical point is strictly preperiodic (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 preperiodic. (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.)

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[]

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

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, the term is used in complex dynamics; a more appropriate one would be Misiurewicz–Thurston points (after William Thurston, who investigated postcritically 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

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

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

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

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.

Properties of Misiurewicz points of complex quadratic mapping[]

Misiurewicz points belong to the boundary of the Mandelbrot set. Misiurewicz points are dense in the boundary of the Mandelbrot set.^[3][4]

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.^[5]

Types[]

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Misiurewicz points can be classified according to the number of external rays that land on them i.e. points where the branches meet:^[3]

• branch points, which disconnect the Mandelbrot set, with 3 or more external arguments (or angles);
• non-branch points, with exactly 2 external arguments (points of arcs within the Mandelbrot set) that are less conspicuous and not easily found in pictures; and
• end points, with 1 external argument (branch tips).

According to the Branch Theorem of the Mandelbrot set,^[4] all branch points of the Mandelbrot set are Misiurewicz points. Also, in a combinatorial sense, hyperbolic components are represented by their centers.^[3][4]

Most Misiurewicz parameters in the Mandelbrot set look like "centers of spirals".^[6] 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 :

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^[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[]

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[]

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

Point ${\displaystyle c=-0.10109636384562...+0.95628651080914...*i=M_{4,1}}$

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

See also[]

• Arithmetic dynamics
• Feigenbaum point
• Dendrite (mathematics)

References[]

Further reading[]

External links[]

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