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fixed point iteration
Here's a general link about iterated functions: Iterated function - Wikipedia, the free encyclopedia
Consider f(x)=x^{2} -2 for x in the closed interval [-2,2]. Let f^{n} be the nth iterate of f. Also let a be in [-2,2]; define the sequence x_{n} by x_{0} = a and x_{n+1} = f(x_{n}).
1. Is there a with x_{n} unequal to -1 for all n, but x_{n} converges to -1?
2. Is the set D={a : x_{n} = 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 ??