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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.

MSC:

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

Citations:

Zbl 0970.22006
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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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