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We study the mean orbital pseudo-metric for Polish dynamical systems and itsconnections with properties of the space of invariant measures. We giveequivalent conditions for when the set of invariant measures generated byperiodic points is dense in the set of ergodic measures and the space ofinvariant measures. We also introduce the concept of asymptotic orbital averageshadowing property and show that it implies that every non-empty compactconnected subset of the space of invariant measures has a generic point.

The mean orbital pseudo-metric and the space of invariant measures