Basins of attraction for Newton's method and escape basins for open Hamiltonian scattering. Fractal boundaries reveal Wada basins — every boundary point is on the boundary of all basins simultaneously.
Newton's method for zⁿ=1 produces fractal basin boundaries. For n≥3, the basins form Wada basins (Yoneyama 1917) — a topological pathology where any open ball at the boundary intersects all basins. Hamiltonian escape basins (Bleher et al. 1988) show similar fractal structure. Basin boundary dimension = 1 + 1/ln(3) ≈ 1.631 for Newton's z³-1.