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#1 2005-12-02 15:21:43
- yttrium88
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Iterated Polynomials
Ok I admit it, this is more of a ponder than a discovery.
First, by ƒ^n(x), I mean the function ƒ iterated on x, n times.
Now, if P(x) is a polynomial, is P^∞(x) a polynomial?
Is it continous?
Is it a step (also called discrete, I think) function?
Is it just a fractal set?
Hmmm....
#2 2005-12-29 06:28:25
- God
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Re: Iterated Polynomials
I would say it is not a polynomial. Apart from the fact that you cannot iterate a polynomial infinitely many times, consider this:
ƒ(x) = x^2
ƒa(0) will always be 0, and ƒa(1) and ƒa(-1) will always be 1. For all x not equal to 0 such that |x| < 1, ƒa(x) approaches 0 as a approaches ∞, and for all other numbers, ƒa(x) tends to ∞ as a tends to ∞ - that is, it does not exist. So what you're left with in a limiting case is a discontinuous (and nonexistent) function defined only for the domain [-1, 1].