Convergence of non-ergodic dynamical systems

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Let {Tt} be a flow on a probability space (S,L,μ}) which describes the time evolution of a dynamical system with state space S, and interpret μ as the initial distribution of the system. Then the distribution of the system at time t is given by μTt−1. Our aim is to study the asymptotic behavior of μTt−1both in general and in the particular cases of random rate and almost periodic systems. The results seem to indicate that convergence or mean convergence is the normal behavior in the non-ergodic case.