## 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.

### MSC:

 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 22D40 Ergodic theory on groups

### Keywords:

isomorphism rigidity; commuting group automorphisms

Zbl 0970.22006
Full Text:

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