Strongly Independent Matrices and Applications on the Rigidity of $A$-Invariant Measures on $n$-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 ${\mathbb{T}}^n$.

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