Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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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].