  We study the system of axion strings that forms in the early Universe if the
Peccei-Quinn symmetry is restored after inflation. Using numerical simulations,
we establish the existence of an asymptotic solution to which the system is
attracted independently of the initial conditions. We study in detail the
properties of this solution, including the average number of strings per Hubble
patch, the distribution of loops and long strings, the way that different types
of radiation are emitted, and the shape of the spectrum of axions produced. We
find clear evidence of logarithmic violations of the scaling properties of the
attractor solution. We also find that, while most of the axions are emitted
with momenta of order Hubble, most of the axion energy density is contained in
axions with energy of order the string core scale, at least in the parameter
range available in the simulation. While such a spectrum would lead to a
negligible number density of relic axions from strings when extrapolated to the
physical parameter region, we show that the presence of small logarithmic
corrections to the spectrum shape could completely alter such a conclusion. A
detailed understanding of the evolution of the axion spectrum is therefore
crucial for a reliable estimate of the relic axion abundance from strings.
