  A nearly-integrable dynamical system has a natural formulation in terms of
actions, $y$ (nearly constant), and angles, $x$ (nearly rigidly rotating with
frequency $\Omega(y)$). We study angle-action maps that are close to symplectic
and have a positive-definite twist, the derivative of the frequency map,
$D\Omega(y)$. When the map is symplectic, Nekhoroshev's theorem implies that
the actions are confined for exponentially long times: the drift is
exponentially small and numerically appears to be diffusive. We show that when
the symplectic condition is relaxed, but the map is still volume-preserving,
the actions can have a strong drift along resonance channels. Averaging theory
is used to compute the drift for the case of rank-$r$ resonances. A comparison
with computations for a generalized Froeschl\'e map in four-dimensions, shows
that this theory gives accurate results for the rank-one case.
