The attractive behaviour of ultra-slow-roll inflation
  It is often claimed that the ultra-slow-roll regime of inflation, where the
dynamics of the inflaton field are friction dominated, is a non-attractor
and/or transient. In this work we carry out a phase-space analysis of
ultra-slow roll in an arbitrary potential, $V(\phi)$. We show that while
standard slow roll is always a dynamical attractor whenever it is a
self-consistent approximation, ultra-slow roll is stable for an inflaton field
rolling down a convex potential with $M_{\scriptscriptstyle{\mathrm{Pl}}}
V''>|V'|$ (or for a field rolling up a concave potential with
$M_{\scriptscriptstyle{\mathrm{Pl}}} V''<-|V'|$). In particular, when
approaching a flat inflection point, ultra-slow roll is always stable and a
large number of $e$-folds may be realised in this regime. However, in
ultra-slow roll, $\dot{\phi}$ is not a unique function of $\phi$ as it is in
slow roll and dependence on initial conditions is retained. We confirm our
analytical results with numerical examples.
