Math Is Fun Forum

yttrium88

Member

Registered: 2005-12-01

Posts: 20

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....

God

Member

Registered: 2005-08-25

Posts: 59

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].