Invariant measures on the n-torus that survive a sparse family of shifted power maps must be finitely supported or Lebesgue, provided the shifts form a strongly independent tuple.
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Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus
Invariant measures on the n-torus that survive a sparse family of shifted power maps must be finitely supported or Lebesgue, provided the shifts form a strongly independent tuple.