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We prove that local rank of an ergodic symmetric power $T^{\odot n}$ does notexceed $n!n^{-n}$. A. Katok's old results show that this upper bound is exact.We prove also that $T^{\odot n}$ has infinite Rank as $n>1$.

Local Rank of Ergodic Symmetric $n$-Powers does not exceed $n!n^{-n}$