  Due to the high stellar densities in young clusters, planetary systems formed
in these environments are likely to have experienced perturbations from
encounters with other stars. We carry out direct $N$-body simulations of
multi-planet systems in star clusters to study the combined effects of stellar
encounters and internal planetary dynamics. These planetary systems eventually
become part of the Galactic field population the parental cluster dissolves,
which is where most presently-known exoplanets are observed. We show that
perturbations induced by stellar encounters lead to distinct signatures in the
field planetary systems, most prominently, the excited orbital inclinations and
eccentricities. Planetary systems that form within the cluster's half-mass
radius are more prone to such perturbations. The orbital elements are most
strongly excited in the outermost orbit, but the effect propagates to the
entire planetary system through secular evolution. Planet ejections may occur
long after a stellar encounter. The surviving planets in these reduced systems
tend to have, on average, higher inclinations and larger eccentricities
compared to systems that were perturbed less strongly. As soon as the parental
star cluster dissolves, external perturbations stop affecting the escaped
planetary systems, and further evolution proceeds on a relaxation time scale.
The outer regions of these ejected planetary systems tend to relax so slowly
that their state carries the memory of their last strong encounter in the star
cluster. Regardless of the stellar density, we observe a robust anticorrelation
between multiplicity and mean inclination/eccentricity. We speculate that the
"Kepler dichotomy" observed in field planetary systems is a natural consequence
of their early evolution in the parental cluster.
