  We examine the low temperature behavior of the mixed state of a layered
superconductor in the vicinity of a quantum critical point separating a pure
superconducting phase from a phase in which a competing order coexists with
superconductivity. At zero temperature, we find that there is an avoided
critical point in the sense that the phase boundary in the limit $B\to 0$ does
not connect to the B=0 critical point. Consequently, there exists a quasi-1D
regime of the phase diagram, in which the competing order is largely confined
to 1D ``halos'' about each vortex core, and in which interactions between
neighboring vortices, although relevant at low temperature, are relatively
weak.
