  In renormalized field theories there are in general one or few fixed points
which are accessible by the renormalization-group flow. They can be identified
from the fixed-point equations. Exceptionally, an infinite family of fixed
points exists, parameterized by a scaling exponent $\zeta$, itself function of
a non-renormalizing parameter. Here we report a different scenario with an
infinite family of fixed points of which seemingly only one is chosen by the
renormalization-group flow. This dynamical selection takes place in systems
with an attractive interaction ${\cal V}(\phi)$, as in standard $\phi^4$
theory, but where the potential $\cal V$ at large $\phi$ goes to zero, as e.g.
the attraction by a defect.
