In what sense (if at all) does the iteration $x \mapsto 2x^2 - 1$ converge to $\cos 2^n x$ in the unit interval [-1,1]?
One might try to plot in Mathematica:
y = x; Table[y = 2 y^2 - 1; y, {k, 1, 5}] Plot[%, {x, -1, 1}]
And you get an some plots resembling cosines of increasing frequency. Can this be made rigorous?
This is related to a similar question I asked.