Quadratic-like maps and Renormalization
Nuria Fagella

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).

[1] Nuria Fagella The theory of polynomial-like mappings - The importance of quadratic polynomials

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updated 30 Dec 2013