Summary
LetT be a weakly mixing transformation with respect to a probability measureP on a metric space (X, d). Suppose further that every open ball of (X, d) has positive measure. Then we show that, for anyP-measurable setA withP(A) > 0, lim supD k (T n A) =D k (X) fork = 2, 3,⋯, whereD k (B) is the geometric diameter of orderk of a subsetB ofX. It is shown further that “D k ” can be replaced by “essD k ”, in the case whenTB is measurable wheneverB is measurable. These results complement a previous one due to R. E. Rice for strongly mixing transformations and improve a result of C. Sempi on weakly mixing transformations.
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Fatkić, H. Note on weakly mixing transformations. Aeq. Math. 43, 38–44 (1992). https://doi.org/10.1007/BF01840473
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DOI: https://doi.org/10.1007/BF01840473