  The finite-frequency dynamics of the $^4$He superfluid phase transition can
be formulated in terms of the response of thermally excited vortex loops to an
oscillating flow field. The key parameter is the Hausdorff fractal dimension
$d_H$ of the loops, which affects the dynamics because the frictional force on
a loop is proportional to the total perimeter $P$ of the loop, which varies as
$P \sim a^{d_H}$ where $a$ is the loop diameter. Solving the 3D Fokker-Planck
equation for the loop response at frequency $\omega $ yields a superfluid
density which varies at $T_{\lambda}$ as $\omega^{1/(d_H -1)}$. This power-law
variation with $\omega$ agrees with the scaling form found by Fisher, Fisher,
and Huse, since the dynamic exponent $z$ is identified as $z = d_H-1$. Flory
scaling for the self-avoiding loops gives a fractal dimension in terms of the
space dimension $d$ as $d_H = (d+2)/2$, yielding $z = d/2 = 3/2$ for d = 3, in
complete agreement with dynamic scaling.
