  Ergodicity of random dynamical systems with a periodic measure is obtained on
a Polish space. In the Markovian case, the idea of Poincar\'e sections is
introduced. It is proved that if the periodic measure is PS-ergodic, then it is
ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only
has equally placed simple eigenvalues including $0$ on the imaginary axis, then
the periodic measure is PS-ergodic and has positive minimum period. Conversely
if the periodic measure with the positive minimum period is PS-mixing, then the
infinitesimal generator only has equally placed simple eigenvalues (infinitely
many) including $0$ on the imaginary axis. Moreover, under the spectral gap
condition, PS-mixing of the periodic measure is proved. The ``equivalence" of
random periodic processes and periodic measures is established. This is a new
class of ergodic random processes. Random periodic paths of stochastic
perturbation of the periodic motion of an ODE is obtained.

Probability