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

Abstract

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