A while ago I asked a question about whether any pair of attracting periodic lengths do occur in a polynomial (

forum link).

With the above article and some modifications, the answer is in the affirmative: For any positive integers n,m one can construct a polynomial with superattracting cycles of those lengths and additionally with any given locations in the complex plane.

Below is an example where I put a square (cyan lines) and a triangle (green) into one polynomial (Julia set in the 11-square, cropped out).

`p(z)=(-185069.98885599966-28779.597150006157i)*z^13+(-320903.2134525415-157919.47351622628i)*z^12+(-315719.3790644852-101990.91826610581i)*z^11+(-276924.36018819432-14896.022791463125i)*z^10+(-163537.04558488802+27255.235582973983i)*z^9+(-69968.277391602838+35371.68830630045i)*z^8+(-17631.574363018117+21441.241058748121i)*z^7+(-1871.7837135907575+9572.4353604984062i)*z^6+(795.77790444995117+1759.8198661876359i)*z^5+(836.36702810849238+258.78509404812041i)*z^4+(106.72887347105063+58.936192381961405i)*z^3+(4.1319822925307754-29.085935503481021i)*z^2+(3.7912920048968379-2.3081860986015941i)*z^1+(-0.42993374393102507-0.069566057852280316i)`

**Method's details**Given 2 desired cycle lengths n,m and a set of complex points a

_{i} with cycles a

_{1}..a

_{n} and a

_{n+1}..a

_{n+m}, the following polynomial shows two super-attracting cycles of lengths n and m at those positions:

\[

H(z) := \left [

\sum\limits_{i=1}^{n-1} a_{i+1} \cdot h_i(z)

+ a_1 \cdot h_n(z)

\right ] + \left [

\sum\limits_{i=n+1}^{n+m-1} a_{i+1} \cdot h_i(z)

+ a_{n+1} \cdot h_{n+m}(z)

\right ]

\]

The first term (in square brackets) exhibits the period n, the 2nd term the cycle of length m.

The function h

_{i}(z) is designed to fulfill the following demands (formula 9 in the article):

- h
_{i}(a_{j}) vanishes at every periodic point (independent of the cycle) other then a_{i}, i.e. h_{i}(a_{ j}) = 1 iff i=j, else 0 - the derivative h
_{i}'(a_{j}) = 0 at every periodic point, so every cycle is superattracting

\[

h_i(z) := [1-2 \cdot l_i'(a_i) \cdot (z-a_i)] \cdot [l_i(z)]²

\]

where l

_{i} is a characteristic function (no demand for the derivative here) for periodic point a

_{i}. The function has roots at every periodic point other then a

_{i}, and at that point the function's value is 1:

\[

l_i(z) := \prod\limits_{j=1 , j \neq i}^{n+m} \frac{(z-a_j)}{(a_i-a_j)}

\]

(I hope I wrote the correct subscripts here in the text. I'll re-check a couple of times in the next days).