>>12974115For the pic,
Let F(x,t) = x + f(x)*t + f(f(x))*t^2 + ...
In the given equation, if you replace x by (f^n)(x) then multiply by t^(n+2) then sum from n = 0 to infinity, you can start putting the F's in.
SUM f^(n+2)*t^(n+2) + a*t*f^(n+1)*t^(n+1) = (t^2)b(a+b) SUM (f^n)*t^n
(F - x - f(x)t) + at(F-x) = (t^2)b(a+b)F
F*(1+at - b(a+b)t^2) = x(1+at) + f(x)*t
F = (x(1+at) + f(x)*t)/(1+at - b(a+b)t^2))
Expanding the rhs as a power series in t (after partial fraction decomposition) gives a relation between f^n and f.
Assume the partial fraction decomposition of 1/(1 + at - b(a+b)t^2) is of the form A/(B-t) - A/(B-t) where A,B,C are functions of a,b.
Depending on B and C, f will be exponentially growing, decaying, or constant and could be oscillating or not.
This is where op's pic assumes replacing x_(n+k) with (x_n)*x^k where x is the growth rate.
They are essentially solving for the growth rate(s) by solving for x.
