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**Zero-entropy algebraic \(\mathbb{Z}^d\)-actions that do not exhibit rigidity.**
*(English)*
Zbl 1018.37004

Isomorphism rigidity (the property that measurable conjugacies must coincide a.e. with an affine map) was shown for irreducible algebraic \(\mathbb{Z}^d\) (\(d>1\)) actions by B. Kitchens and K. Schmidt [Invent. Math. 142, 559-577 (2000; Zbl 0970.22006)]. Irreducibility implies that all \(\mathbb{Z}^2\)-subactions have zero entropy, and they raised the question of whether isomorphism rigidity extends to zero-entropy \(\mathbb{Z}^d\)-actions of higher entropy rank (in which there are \(\mathbb{Z}^r\)-subactions of positive entropy for some \(r\), \(1<r<d\)). In this important contribution, an example of a \(\mathbb{Z}^8\)-action is found that does not exhibit isomorphism rigidity.

This is far from the end of the story. In subsequent work by the same author, a weaker form of rigidity is recovered for zero entropy \(\mathbb{Z}^d\)-actions on zero-dimensional groups: measurable conjugacies coincide a.e. with a ‘polynomial map’ whose properties are governed by the entropy rank, and in particular measurable conjugacies must be continuous.

This is far from the end of the story. In subsequent work by the same author, a weaker form of rigidity is recovered for zero entropy \(\mathbb{Z}^d\)-actions on zero-dimensional groups: measurable conjugacies coincide a.e. with a ‘polynomial map’ whose properties are governed by the entropy rank, and in particular measurable conjugacies must be continuous.

Reviewer: Thomas Ward (Norwich)

### MSC:

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |

22D40 | Ergodic theory on groups |

### Citations:

Zbl 0970.22006### References:

[1] | A. Katok, S. Katok, and K. Schmidt, Rigidity of measurable structure for \(\mathbbZ^d\)-actions by automorphisms of a torus , · Zbl 1035.37005 · doi:10.1007/PL00012439 |

[2] | B. Kitchens and K. Schmidt, “Markov subgroups of \((\Z/2\Z)^\Z^2\)” in Symbolic Dynamics and Its Applications (New Haven, Conn., 1991) , Contemp. Math. 135 , Amer. Math. Soc., Providence, 1992, 265–283. |

[3] | –. –. –. –., Isomorphism rigidity of irreducible algebraic \(\mathbbZ^d\)-actions , Invent. Math. 142 (2000), 559–577. · Zbl 0970.22006 · doi:10.1007/s002220000098 |

[4] | D. Lind, K. Schmidt, and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups , Invent. Math. 101 (1990), 593–629. · Zbl 0774.22002 · doi:10.1007/BF01231517 |

[5] | A. N. Quas and P. B. Trow, Mappings of group shifts , Israel J. Math. 124 (2001), 333–365. · Zbl 1035.37012 · doi:10.1007/BF02772629 |

[6] | D. J. Rudolph and K. Schmidt, Almost block independence and Bernoullicity of \(\mathbbZ^d\)-actions by automorphisms of compact abelian groups , Invent. Math. 120 (1995), 455–488. · Zbl 0835.28007 · doi:10.1007/BF01241139 |

[7] | M. A. Shereshevsky, On the classification of some two-dimensional Markov shifts with group structure , Ergodic Theory Dynam. Systems 12 (1992), 823–833. · Zbl 0781.58017 · doi:10.1017/S0143385700007124 |

[8] | K. Schmidt, Dynamical Systems of Algebraic Origin , Progr. Math. 128 , Birkhäuser, Basel, 1995. · Zbl 0833.28001 |

[9] | –. –. –. –., “The dynamics of algebraic \(\mathbbZ^d\)-actions” in European Congress of Mathematics (Barcelona, 2000), Vol. 1 , Progr. Math. 201 , Birkhäuser, Basel, 2001, 543–553. \CMP1 905 342 · Zbl 1071.28011 |

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