  We prove a variant of the Chance-McDuff conjecture for pseudo-rotations:
under certain additional conditions, a closed symplectic manifold which admits
a Hamiltonian pseudo-rotation must have deformed quantum product and, in
particular, some non-zero Gromov-Witten invariants. The only assumptions on the
manifold are that it is weakly monotone and that its minimal Chern number is
greater than one. The conditions on the pseudo-rotation are expressed in terms
of the linearized flow at one of the fixed points and hypothetically satisfied
for most (but not all) pseudo-rotations.
