  Eternal inflation is a seemingly generic consequence of theories that give
rise to accelerated expansion of the universe and possess multiple vacuum
states. Making predictions in an eternally inflating universe is notoriously
difficult because one must compare infinite quantities, and a wide variety of
regulating procedures yield radically different results. This is the measure
problem of eternal inflation. In this paper, we analyze models of eternal
inflation which allow for the possibility of cyclic bubble universes: in each
bubble, standard cosmological evolution is re-played over and over again.
Eternal inflation can generically arise in cyclic models that include a dark
energy dominated phase. In such models, several problematic consequences of
standard regulating procedures, such as the youngness and Boltzmann Brain
problems, are substantially alleviated. We discuss the implications for making
predictions in cyclic models, as well as some general implications for
understanding the measure problem in eternal inflation.
