6.2: Misiurewicz points and the M-set self-similarity
- Page ID
- 102618
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Here are illustrations of M near some of Misiurewicz points. Preperiodic points are in the center of the pictures. The images are zoomed 4, 3 and 1.3283 = 2.34 times respectevely. Some self-similar periodic points with its period are shown too.



In the last figures rotational angle is very close to 120o, which accounts for the 3-fold rotational symmetry in the picture. In the center of the picture one has 3 lines meeting, and there are numerous nearby points where 6 lines meet. At each of the letter points there is a tiny replica of M.
You see, that preperiodic points explain too spokes symmetry in the largest antenna attached to a primary bulb.
These pictures have next features in common:
- The preperiodic points are not in black regions of M.
- They exhibit self-similarity, i.e., they look roughly the same at shrinking the picture centered at preperiodic point by a factor of |λ| and rotating through the angle Arg(λ). This becomes more precise as the magnification increases. The rotational angles of the sequences are -23.1256o and 119.553o respectively. This accounts for the slight changes in orientation under successive magnifications in figures.
- There is a sequence of miniature Ms of decreasing size converging to the point. Each of them has a periodic point in its main cardioid (see the theorem below). When we shrink the picture by a factor of λ the miniature Ms shrink by a factor of λ2 therefore nearby miniature Ms shrink faster than the view window, so they eventually disappear.
- There is a fourth feature not visible in these pictures: For preperiodic point co , the Julia set J(co ) near the point z = co looks very much like the Mandelbrot set M near co . This is a theorem of Lei, which we will discuss on the next page.
Theorem Let co be a preperiodic point with period 1. Let cn denote the nearest periodic point with period n. Then as n approaches infinity
(cn - c0 )/(cn+1 - c0 ) → λ = 2h
where h is the fixed point of the critical orbit of co .
Proof: We will use Newton's approximation to find a root of an equation fCnon(0) = 0 for periodic point cn with period n near co . If (cn - co) value is small enough, then
fCnon(0) = fCo+(Cn-Co)on(0) = fCoon(0) + (cn - co) d/dc fCon(0) |C=Co = 0
(we do not prove that we can use this approximation). For simplicity we will denote
dn = d/dc fCon(0) |C=Co .
As co is preperiodic with period 1, than fCoon(0) = h for large enough n, therefore
cn - co = - h/dn .
Since fCo(n+1)(0) = [fCon(0)]2 + c, it follows that for large n
dn+1 = 2 h dn + 1
and
(cn - co )/(cn+1 - co ) = (h/dn)/(h/dn+1) = dn+1 /dn = 2h + 1/dn.
The limit of this as n approaches infinity is 2h as claimed, because dn gets arbitrarily large for large n.