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

TL;DR

This paper establishes an upper bound on the local rank of ergodic symmetric powers of transformations, confirms its optimality, and shows that these powers have infinite rank for n>1.

Contribution

It provides a precise upper bound for the local rank of ergodic symmetric powers and proves their infinite rank for all n greater than 1.

Findings

Local rank of ergodic symmetric powers does not exceed n!n^{-n}

The upper bound is proven to be exact by previous results

Symmetric powers have infinite rank for n>1

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 $n>1$.