Newton–Lyapunov manifold · basins of z^n − 1
Ask a computer to solve z^d = 1 by Newton's method and colour every starting point by which root it finally reaches. The borders between the basins are never smooth: arbitrarily close to any boundary point sit starting values that run off to all d roots. That infinitely mixed boundary is the same structure that makes weather prediction lose skill after a few days.
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