Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus

Abstract

We introduce the concept of strongly independent matrices over any field, and prove the existence of such matrices for certain fields and the non-existence for algebraically closed fields. Then we apply strongly independent matrices over rational numbers to obtain a measure rigidity result for endomorphisms on $n$-torus.

The paper is rewritten. New theorems are added. 15 pages. Comments welcome