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Isomorphism and Embedding of Borel Systems on Full Sets

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An Erratum to this article was published on 27 September 2013

Abstract

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t.

As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.

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Notes

  1. A notion with the same name and similar (but not identical) definition was introduced earlier by R. Bowen [3].

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Acknowledgements

My thanks to Mike Boyle for some very interesting discussions and many useful suggestions on the presentation of this paper. This work was supported by NSF grant 0901534.

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Correspondence to Michael Hochman.

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Research supported by NSF grant 0901534.

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Hochman, M. Isomorphism and Embedding of Borel Systems on Full Sets. Acta Appl Math 126, 187–201 (2013). https://doi.org/10.1007/s10440-013-9813-8

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