Isomorphism Problem Revisited: Information Spectrum Approach

Abstract

The isomorphism problem in the ergodic theory is revisited from the perspective of information spectrum approach, an approach that has been developed to investigate coding problems for non-ergodic random processes in information theory. It is proved that the information spectrum is invariant under isomorphisms. This result together with an analysis of information spectrum provide a conceptually simple proof of the result by \v{S}ujan, which claims that the entropy spectrum is invariant under isomorphisms. It is also discussed under what circumstances the same information spectrum implies the existence of an isomorphism.