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Chaos of Transformations Induced Onto the Space of Probability Measures

For a dynamical system (X,T), let (M(X),TM) be its induced dynamical system on the space of Borel probability measures with weak*-topology. It is proved that (M(X),TM) is a-transitive (resp., exact, uniformly rigid) if and only if (X,T) is weakly mixing and a-transitive (resp., exact, uniformly rigid), where a is an n-vector of integers. Moreover, some analogous results are obtained for the hyperspace.