  We study the rate of convergence of Wilkinson's shift iteration acting on
Jacobi matrices with simple spectrum. We show that for AP-free spectra (i.e.,
simple spectra containing no arithmetic progression with 3 terms), convergence
is cubic. In order 3, there exists a tridiagonal symmetric matrix P_0 which is
the limit of a sequence of a Wilkinson iteration, with the additional property
that all iterations converging to P_0 are strictly quadratic. Among tridiagonal
matrices near P_0, the set X of initial conditions with convergence to P_0 is
rather thin: it is a union of disjoint arcs X_s meeting at P_0, where s ranges
over the Cantor set of sign sequences s: N -> {1,-1}. Wilkinson's step takes
X_s to X_{s'}, where s' is the left shift of s. Among tridiagonal matrices
conjugate to P_0, initial conditions near P_0 but not in X converge at a cubic
rate.
