# How can I find parameter of the rational map?

I have a family of rational maps $$f$$ ( the Blaschke fraction ) with one complex parameter $$\rho$$:

$$f(z) = \rho z^2 \frac{z-3}{1-3z}$$

I want to find $$\rho$$ such that map $$f$$ has a parabolic period 3 cycle on the unit circle.

So $$\rho$$ should be a solution of system of equations:

$$\begin{cases}f^3(z_0) = z_0 \\ |(f^3)'(z_0)| = 1 \end{cases}$$

where

• $$f^3(z) = f(f(f(z)0)))$$ is 3-rd iteration of function f
• $$(f^3)'(z_0)$$ is the first derivative of $$f^3$$ with respect to $$z$$ at $$z_0$$
• $$z_0$$ is the point from period 3 parabolic cycle

The image of dynamic plane with Julia set:

I know that $$\rho$$ has a modulus equal to one (complex number):

$$|\rho| = 1$$

My first try was easy solution ( point on the unit circle with angle in turns = 1/3)

$$\rho = e^{2 \pi i / 3} = -0.5 +0.8660254037844386 i$$

Then the period 3 cycle is on the unit cirlce but the cycle is not parabolic and the image is different.

Q. How can I find such $$\rho$$ using algebra or numerical root finding ? How can I solve above system of equations ?

• does "a parabolic 3-cycle on the unit circle" mean $|\rho|=1$ or $|z_0|=1$ (or both?)? Jun 18, 2022 at 14:32
• @Claude Good question. I do not know the answer.
Jun 20, 2022 at 16:18

It is an approximation of the answer using brute force and visual analysis.

First analyze the map f

• find critical points
• compute Mandelbrot set
• find parameter $$\rho$$ : point on the unit circle which is parabolic point : cusp or a root

So there are 3 critical points :

• 2 finite critical points : z=1.0 i z= 0.0
• infinity

Here is parameter plane ( rho plane)

Here zoom around cusp point ( black) on the unit circle ( white)

So $$\rho \approx -0.6170144002709304 +0.7869518599370003*I$$

There are 3 Fatou components:

• (white = 255) basin of attraction to infinity
• ( light gray = 195) basin of attraction to fixed point z=0 ( superattracting) with critical point z=0
• (dark gray = 135) basin of parabolic period 3 cycle ( critical point z= 1)

Increasing precision of $$\rho$$ value : there are two 3 point cycle. Choose rho such that 2 cycles join into one cycle.

$$\rho = e^{2 \pi i t}$$

t is changing from 0.3558289901601067 to 0.3558800000000000

Final rho (t) = e^(2piti) is rho = -0.6172665900123702 +0.7867540637673888I with turn(rho) =0.3558800000000000 and radius = 1.0000000000000000

On the last image critical orbit goss to limit cycle from both sides . I think that it is numerical error . Rho value seem to be good