  We study the renormalization flow of generic actions that depend on the
invariants of the field strength tensor of an abelian gauge field. While the
Maxwell action defines a Gaussian fixed point, we search for further
non-Gaussian fixed points or rather fixed functions, i.e., globally existing
Lagrangians of the invariants. Using standard small-field expansion techniques
for the resulting functional flow equation, a large number of fixed points is
obtained, which - in analogy to recent findings for a shift-symmetric scalar
field - we consider as approximation artifacts. For the construction of a
globally existing fixed function, we pay attention to the use of proper initial
conditions. Parametrizing the latter by the photon anomalous dimension, both
the coefficients of the weak-field expansion are fully determined and those of
the large-field expansion can be matched such that a global fixed function can
be constructed for magnetic fields. The anomalous dimension also governs the
strong-field limit. Our results provide evidence for the existence of a
continuum of non-Gaussian fixed points parametrized by a small positive
anomalous dimension below a critical value. We discuss the implications of this
result within various scenarios with and without additional matter. For the
strong-field limit of the 1PI QED effective action, where the anomalous
dimension is determined by electronic fluctuations, our result suggests the
existence of a singularity free strong-field limit, circumventing the standard
conclusions connected to the perturbative Landau pole.
