[seqfan] Re: Connection between Galton-Watson branching process and Julia sets?

israel at math.ubc.ca israel at math.ubc.ca
Thu Jun 2 08:58:13 CEST 2016

If u(n) = z(n) - 1/2, this recurrence becomes u(n) = u(n-1)^2 - u(n-1). 
With u(0) = t, u(n) is a polynomial in t of degree 2^n with integer 
coefficients: u(1) = t^2-t u(2) = t^4-2*t^3+t u(3) = 
t^8-4*t^7+4*t^6+2*t^5-5*t^4+2*t^3+t^2-t u(4) = 
u(5) = 
The table of coefficients of this doesn't seem to be in the OEIS. Maybe it 
should be.


On Jun 1 2016, Alonso Del Arte wrote:

>Given z_0 = i = sqrt(-1), z_n = z_(n - 1)^2 + 1/4, we get a sequence of all
>real numbers (aside from z_0) and all positive numbers (aside from z_0 and
>z_1): i, -3/4, 13/16, 233/256, 70673/65536, etc. Obviously the denominators
>are powers of 4. My first thought for the numerators was Fibonacci numbers,
>but the coincidence is broken by 70673. Just the first four real numerators
>are enough to bring up A015701, which involves iteration in the
>Galton-Watson branching process.
>The mention of iteration suggests I got the right match, but I had never
>heard of Galton-Watson before. From what I've read in the second Google
>result http://galton.uchicago.edu/~lalley/Courses/312/Branching.pdf , it
>makes sense that the denominators would be powers of 4 as well. But the
>OEIS entry says nothing about fractals or at least fractions. I've been
>scrutinizing the Julia sets for quadratic functions with c purely real.
>Any thoughts?

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