  We revisit the scaling properties of the resistivity and the current-voltage
characteristics at and below the Berezinskii-Kosterlitz-Thouless transition,
both in zero and nonzero magnetic field. The scaling properties are derived by
integrating the renormalization group flow equations up to a scale where they
can be reliably matched to simple analytic expressions. The vortex fugacity
turns out to be dangerously irrelevant for these quantities below Tc, thereby
altering the scaling behavior. We derive the possible crossover effects as the
current, magnetic field, or system size is varied, and find a strong
multiplicative logarithmic correction near Tc, all of which is necessary to
account for when interpreting experiments and simulation data. Our analysis
clarifies a longstanding discrepancy between the finite size dependence found
in many simulations and the current-voltage characteristics of experiments. We
further show that the logarithmic correction can be avoided by approaching the
transition in a magnetic field, thereby simplifying the scaling analysis. We
confirm our results by large-scale numerical simulations, and calculate the
dynamic critical exponent z, for relaxational Langevin dynamics and for
resistively and capacitively shunted Josephson junction dynamics.
