Zero-entropy algebraic $\mathbb {Z}\sp d$-actions that do not exhibit rigidity

Abstract

We show that there exist mixing zero-entropy algebraic $\mathbb {Z}\sp 8$-actions that are measurably and topologically conjugate but not algebraically conjugate. This result gives the first known examples of mixing zero-entropy algebraic $\mathbb {Z}\sp d$-actions that do not have rigidity properties, and it provides a negative answer to the isomorphism rigidity problem raised in B. Kitchens and K. Schmidt (Isomorphism rigidity of irreducible algebraic $\mathbb {Z}\sp d$-actions, Invent. Math. 142 (2000), 559‒577) and Schmidt ("The dynamics of algebraic $\mathbb {Z}\sp d$-actions" in European Congress of Mathematics (Barcelona, 2000), Vol. 1, Progr. Math. 201, Birkhäuser, Basel, 2001, 543‒553).