Example D Sometimes a polynomial-like map is created as some iterate of
a function restricted to a domain. For example, for Q_{c}(z) =
z^{2} + c, c_{o} ~ -1.75488 and
U' = { |Im(z)| < 0.2, |Re(z)| < 0.2}
the polynomial Q_{Co}^{o3} maps U' onto a
larger set U with degree 2. The triple
( Q_{Co}^{o3}|_{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 polinomial-like map of the same or lower degree.
You see below the Mandelbrot set and a magification of its homeomorphic copy near c_{o}.
Example E For c = -1.401155... the map Q_{c} is
the Feigenbaum polynomial, that is the limit of the cascade of period doublings
in the real axis. For any n the polynomial
Q_{c}^{o2n} is renormalizable and all these
renormalizations are hybrid equivalent to itself.
Renormalization of Q_{c}^{o2} 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 Q_{c}^{o2} is renormalizable, z = 0 is fixed after two iterations of the renormalized map. Hence, the renormilized filled Julia set is hybrid equivalent to z^{2} - 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