Strongly Independent Matrices and Applications on the Rigidity of $A$-Invariant Measures on $n$-Torus

TL;DR

This paper introduces strongly independent matrices in integer general linear groups and applies this concept to establish measure rigidity results for endomorphisms on n-dimensional tori, especially when 2n+1 is prime.

Contribution

It defines strongly independent matrices and proves their existence under certain prime conditions, then applies this to measure rigidity on n-torus.

Findings

Existence of strongly independent matrices in GL(n,Z) when 2n+1 is prime.

Strong independence leads to measure rigidity results for toral endomorphisms.

Provides new tools for understanding dynamics on the n-torus.

Abstract

We introduce the notion of strongly independent matrices and show the existence of strongly independent matrices in $GL(n,\mathbb{Z})$ over $\mathbb{Z}^n\setminus\{0\}$ when $2n+1$ is a prime number. As an application of strong independence, we give a measure rigidity result for endomorphisms on $n$-torus $\mathbb{T}^n$.