Entropy

In Section I.2 we introduced one of the fundamental problems of Ergodic Theory, namely, deciding when two automorphisms T₁, T₂ of probability spaces (X₁ , A₁, μ₁) and (X₂, A₂, μ₂)are equivalent. The approach developed there, involving the study of spectral properties of the associated isometric operators $${U_{{T_i}}}$$: L2 (X _i , A _i , μ _i ) → L2 (X _i , A _i , μ _i ) (i = 1,2), led to the concept of spectral equivalence. We proved that equivalent maps are spectrally equivalent, and mentioned that the converse is false.