Isomorphism and embedding of Borel systems on full sets
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 there exists a unique free Borel system (Y,S) which is strictly tuniversal 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 tuniversal system from mixing shifts of finite type of entropy at least 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 positiverecurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiomA diffeomorphisms. In particular any two equalentropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.
 Publication:

arXiv eprints
 Pub Date:
 August 2010
 DOI:
 10.48550/arXiv.1008.3549
 arXiv:
 arXiv:1008.3549
 Bibcode:
 2010arXiv1008.3549H
 Keywords:

 Mathematics  Dynamical Systems;
 37A35
 EPrint:
 17 pages, v2: correction to bibliography