Isomorphism and embedding of Borel systems on full sets

Michael Hochman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


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.

Original languageAmerican English
Pages (from-to)187-201
Number of pages15
JournalActa Applicandae Mathematicae
Issue number1
StatePublished - Aug 2013
Externally publishedYes

Bibliographical note

Funding Information:
Research supported by NSF grant 0901534.


  • Borel dynamics
  • Entropy conjugacy
  • Isomorphism of dynamical systems


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