  We study the resonance (or Gamow) eigenstates of open chaotic systems in the
semiclassical limit, distinguishing between left and right eigenstates of the
non-unitary quantum propagator, and also between short-lived and long-lived
states. The long-lived left (right) eigenstates are shown to concentrate as
$\hbar\to 0$ on the forward (backward) trapped set of the classical dynamics.
The limit of a sequence of eigenstates $\{\psi(\hbar)\}_{\hbar\to 0}$ is found
to exhibit a remarkably rich structure in phase space that depends on the
corresponding limiting decay rate. These results are illustrated for the open
baker map, for which the probability density in position space is observed to
have self-similarity properties.
