 Unit-modulus multipliers do not determine nonlinear stability

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.