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.


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


Zbl 0970.22006
Full Text: DOI


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