5.5: Renormalization
- Page ID
- 102239
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Quadratic-like maps and Renormalization
Example D Sometimes a polynomial-like map is created as some iterate of a function restricted to a domain. For example, for Qc(z) = z2 + c, co ~ -1.75488 and
U' = { |Im(z)| < 0.2, |Re(z)| < 0.2}
the polynomial QCoo3 maps U' onto a larger set U with degree 2. The triple ( QCoo3|U' , U', U ) is a polynomial-like map of degree two (or quadratic-like map).
A polynomial is renormalizable if restriction of some of its iterate gives a polynomial-like map of the same or lower degree.
You see below the Mandelbrot set and a magification of its homeomorphic copy near co.
For periodic point c0 = -1.75488 with period 3 (see "airplane" below) the critical point is fixed under iterations of Qc0o3 therefore the filled Julia set of the quadratic-like map is homeomorphic to circle.
For periodic point c1 = -1.77289 with period 6 we have Qc1o6(0) = 0. In this case Qc1o3 and Qc1o6 are renormalizable. The critical point is periodic of period two under iterations of Qc1o3 therefore the filled Julia set of the quadratic-like map is homeomorphic to the Julia set z2 - 1 (in square). For Qc1o6 the critical point is fixed so the renormalized polynomial is z2 (the greatest bulb in the center)
Example E For c = -1.401155... the map Qc is the Feigenbaum polynomial, that is the limit of the cascade of period doublings in the real axis. For any n the polynomial Qco2n is renormalizable and all these renormalizations are hybrid equivalent to itself.
Renormalization of Qco2 is shown to the left and below.
Example F Let c = 0.419643 + 0.60629i is a Misiurewicz point in the boundary of the Mandelbrot set. For this map z = 0 becomes periodic of period two after three iterations (see the picture). Since Qco2 is renormalizable, z = 0 is fixed after two iterations of the renormalized map. Hence, the renormilized filled Julia set is hybrid equivalent to z2 - 2 , i.e. a quasiconformal image of the interval [-2, 2] (curve 2-0-4 to the left).