Misiurewicz point

Mathematical notation

A parameter c is a Misiurewicz point M_{k,n} if it satisfies the equations: :f_c^{(k)}(z_{cr}) = f_c^{(k+n)}(z_{cr})

and: :f_c^{(k-1)}(z_{cr}) \neq f_c^{(k+n-1)}(z_{cr})

so: :M_{k,n} = c : f_c^{(k)}(z_{cr}) = f_c^{(k+n)}(z_{cr})

where:

• z_{cr} is a critical point of f_c,
• k and n are positive integers,
• f_c^{(k)} denotes the k-th iterate of 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. 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 P_c(z)=z^2+c which has a single critical point at z = 0. The Misiurewicz points of this family of maps are roots of the equations:

: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
• P_c^{(n)} = P_c ( P_c^{(n-1)}) denotes the n-fold composition of P_c(z)= z^2+c with itself i.e. the nth iteration of P_c.

For example, the Misiurewicz points with k= 2 and n= 1, denoted by M2,1, are roots of:

:\begin{align} & P_c^{(2)}(0) = P_c^{(3)}(0)\ \Rightarrow {} & c^2+c = (c^2+c)^2+c \ \Rightarrow {} & c^4+2c^3 = 0. \end{align}

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.

If 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 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 J_c are locally asymptotically self-similar around Misiurewicz points.

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. According to the Branch Theorem of the Mandelbrot set, all branch points of the Mandelbrot set are Misiurewicz points.

Most Misiurewicz parameters within the Mandelbrot set exhibit a "center of a spiral". 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 M_{4,1} 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.

External arguments

External arguments of Misiurewicz points, measured in turns are:

• Rational numbers
• Proper fractions with an even denominator
  • Dyadic fractions with denominator = 2^b and finite (terminating) expansion:\frac{1}{2}{10} = 0.5{10} = 0.1_2
  • Fractions with a denominator = a \cdot 2^b; a, b \in \mathbb{N}; 2 \nmid b and repeating expansion: \frac{1}{6}{10} = {\frac{1}{2 \times 3}}{10}=0.16666..._{10} = 0.0(01)..._2.

The subscript number in each of these expressions is the base of the numeral system being used.

Examples of Misiurewicz points of complex quadratic mapping

End points

Point c = M_{2,1} = -2 is considered an end point as it is the tip of the main antenna of the Mandelbrot set. 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 { 0 , -2, 2, 2, 2, ... }, following the Symbolic sequence = C L R R R ... with a pre-period of 2 and period of 1.

Point c = M_{2,2} = i is considered an end point as it is the tip of one of two main branches of the 1/3 limb, and the landing point of the external ray for the angle 1/6. Its critical orbit is {0, i, i-1, -i, i-1, -i...}.

Point c = -0.228155493653962... + i , 1.115142508039937... = M_{3,1} is the tip of the other main branch of the 1/3 limb. Like all other end points, it is the landing point of only one external ray. It has a pre-period of 3 and a period of 1.

Branch points

Point 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 is the landing point of 3 external rays: 9/56, 11/56 and 15/56. It has a pre-period of 4 and a period of 1.

Other points

Point c = -1.54368901269208... = M_{3,1} is a principal Misiurewicz point of the main antenna of the Mandelbrot set. It is the landing point for two external rays: \frac{5}{12}, \frac{7}{12}, and has pre-period k = 3 and period n = 1.

Point c = -0.77568377... + i , 0.13646737... = M_{23,2} may be recognized as the center of a two-arms spiral located in the so-called "Seahorse Valley" of the Mandelbrot set. It is the landing point of 2 external rays with angles: \frac{8388611}{25165824} and \frac{8388613}{25165824} where the denominator is 3*2^{23}, and has pre-period k = 23 and period n = 2.

References

References

1. (19 January 2022). "Logistic map trajectory distributions:Renormalization-group, entropy, and criticality at the transition to chaos". Chaos 31, 033112 (2021).
2. [http://www.math.iupui.edu/~mmisiure/ Michał Misiurewicz home page], Indiana University-Purdue University Indianapolis
3. Wellington de Melo, Sebastian van Strien, "One-dimensional dynamics". Monograph, Springer Verlag (1991)
4. Adrien Douady, John Hubbard, "Etude dynamique des polynômes complexes", prépublications mathématiques d'Orsay, 1982/1984
5. Dierk Schleicher, "On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets", in: M. Lapidus, M. van Frankenhuysen (eds): Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proceedings of Symposia in Pure Mathematics 72, American Mathematical Society (2004), 477–507 or [https://arxiv.org/abs/math/9902155 online paper from arXiv.org]
6. [http://projecteuclid.org/euclid.cmp/1104201823 Lei.pdf] [[Tan Lei]], "Similarity between the Mandelbrot set and Julia Sets", Communications in Mathematical Physics 134 (1990), pp. 587-617.
7. Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more [[external ray. 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.[https://users.math.yale.edu/public%20html/People/frame/Fractals/MandelSet/MandelBoundary/Mis.html Fractal Geometry Yale University Michael Frame, Benoit Mandelbrot (1924–2010), and Nial Neger November 6, 2022]
8. [http://classes.yale.edu/fractals/MandelSet/MandelBoundary/Mis.html The boundary of the Mandelbrot set] {{Webarchive. link. (2003-03-28 by Michael Frame, Benoit Mandelbrot, and Nial Neger)
9. [http://math.unipa.it/~grim/21_project/CiechYeung.pdf Binary Decimal Numbers and Decimal Numbers Other Than Base Ten by Thomas Kim-wai Yeung and Eric Kin-keung Poon]
10. [http://www.mrob.com/pub/muency/tip.html tip of main antennae by Robert P. Munafo]
11. [http://www.mrob.com/pub/muency/terminalpoint.html Tip of the filaments by Robert P. Munafo]
12. [http://www.ibiblio.org/e-notes/MSet/cperiod.htm Preperiodic (Misiurewicz) points in the Mandelbrot set] by Evgeny Demidov