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 $��$, 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 ${\cal V}(��)$, as in standard $��^4$ theory, but where the potential $\cal V$ at large $��$ goes to zero, as e.g. the attraction by a defect.

v2: Several misprints corrected, appendix on toy model rendered more relevant. 13 pages, 22 figures