  We study smooth, caustic-free, chaotic semiclassical dynamics on
two-dimensional phase space and find that the dynamics can be approached by an
iterative procedure which constructs an approximation to the exact long-time
semiclassical propagator. Semiclassical propagation all the way to the
Heisenberg time, where individual eigenstates are resolved, can be computed in
polynomial time, obviating the need to sum over an exponentially large number
of classical paths. At long times, the dynamics becomes quantum-like, given by
a matrix of the same dimension as the quantum propagator. This matrix, however,
differs both from the quantum and the one-step semiclassical propagators,
allowing for the study of the breakdown of the semiclassical approximation. The
results shed light on the accuracy of the Gutzwiller trace formula in two
dimensions, and on the source of long-time periodic orbit correlations.
