6.3: The M and J-sets similarity, Lei's theorem
- Page ID
- 102619
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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 Julia sets J(co) associated with the preperiodic points co near the point z = co are shown. These pictures are compared with the corresponding areas of the Mandelbrot set.




J pictures differ from the corresponding M pictures in that there are no black regions in them. On the other hand if we zoom in to a preperiodic point in M, the nearby miniature Ms shrink faster than the view window (if we shrink the picture by a factor of m the miniature Ms shrink by a factor of m2), so they eventually disappear.
This local similarity between the Mandelbrot set near a preperiodic point co and the Julia set J(co ) near z = co shown above is the subject a theorem of Tan Lei.
Here is a partial explanation for it in the case when period of co is 1. For small ε
fCo+εo(n+1)(0) = fCo+εon(co + ε)
= fCoon(co ) + ( d/dc fCon(0) |C=Co + d/dz fCoon(z) |z=Co ) ε + O(ε2)
= fCoon(co + knε) + O(ε2) ,
where
kn = ( d/dc fCon(co ) |C=Co + d/dz fCoon(z) |z=Co ) / d/dz fCoon(z) |z=Co .
As since
d/dc fCo(n+1)(co ) |C=Co = 2 hn d/dc fCon(co ) |C=Co + 1 ,
d/dz fCoo(n+1)(z) |z=Co = 2 hn d/dz fCoon(z) |z=Co , hn = fCoon(co )
and hn go to the fixed point h of the critical orbit of preperiodic point co for large enough n, then it can be shown, that kn converge to a finite k [Ravenel].
Equation
fCo+εo(n+1)(0) = fCoon(co + kε) + O(ε2)
means that for small ε the (n+1)th point in critical orbit of c = co+ε can be approximated by the nth point in the Julia orbit of z∗ = co+kε . I.e. the critical orbit is bounded if and only if the z∗ orbit is bounded. This accounts for the local similarity between the Mandelbrot set near co and the Julia set J(co ) near z = co .