This thread was prompted by thread
fixed point iteration

Here's a general link about iterated functions: Iterated function - Wikipedia, the free encyclopedia

Consider f(x)=x2 -2 for x in the closed interval [-2,2]. Let fn be the nth iterate of f. Also let a be in [-2,2]; define the sequence xn by x0 = a and xn+1 = f(xn).

1. Is there a with xn unequal to -1 for all n, but xn converges to -1?
2. Is the set D={a : xn = 2 for some n} dense in [-2,2]?
3. Are there orbits of f of arbitrary length?

I know very little about iterated function theory, so maybe these are "easy" questions, but ??