  We study the collinear fragmentation of highly energetic jets defined with a
small jet radius. In particular, we investigate how the corresponding
fragmentation functions differ from their hadronic counterpart defined in the
common $\overline{\rm MS}$ scheme. We find that the anomalous dimensions
governing the perturbative evolution of the two fragmentation functions differ
starting at the two loop order. We compute for the first time the new anomalous
dimensions at two loops and confirm our predictions by comparing the inclusive
small-$R$ jet spectrum against a fixed order perturbative calculation at ${\cal
O}(\alpha_s^2)$. To investigate the dependence of the anomalous dimension on
the kinematic cutoff variable, we study the fragmentation functions of
Cambridge jets defined with a transverse momentum cutoff as opposed to an
angular cutoff $R$. We further study the evolution of the small-$R$
fragmentation function with an alternative cutoff scale, proportional to $z R$,
representing the maximum possible transverse momentum of emissions within a
jet. In these cases we find that the two-loop anomalous dimensions coincide
with the $\overline{\rm MS}$ DGLAP ones, highlighting a correspondence between
the $\overline{\rm MS}$ scheme and a transverse-momentum cutoff.
