= Unit-modulus multipliers do not determine nonlinear stability
{title2=$DF_\pm(0)=R_\theta,\quad F_\pm(z)=e^{i\theta}(1\pm|z|^2)z$}
For a <discrete dynamical system>, eigenvalues of the linearization strictly inside the unit circle imply local asymptotic stability; an eigenvalue outside implies instability. Eigenvalues on the unit circle require more information. A nontrivial <Jordan block> there can already make the linearized iterates unbounded.
Even a bounded linearization leaves nonlinear stability undecided. The two displayed maps of the complex plane have the same derivative, a <rotation matrix>. Their radial updates are $r\mapsto r(1-r^2)$ and $r\mapsto r(1+r^2)$. For small positive radius, the first decreases to zero and the second moves out of every sufficiently small fixed neighborhood. Thus identical unit-modulus linear multipliers can accompany attraction or instability.
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