Entropy and approximation of measure preserving transformations
T. Schwartzbauer · Pacific Journal of Mathematics · 1972
The relationship between the entropy and rate of approximation of an automorphism was first discovered by A. B. Eatok. He defined for each automorphism T an invariant c(T) which depends only on the rate of approximation of T and then proved that h(T) Sc(T)^ 2h(T) for any ergodic automorphism T 9 where h(T) denotes the entropy of T.The proof which he gave that c(T) ^ 2h(T) can be generalized to the case where T is not ergodic, and it was asserted further that c(T) = 2h(T) if T were ergodic, but the proof given was incomplete.In this paper these results are generalized to the case of an arbitrary automorphism T.We will extend the result c(Γ) = 2fe(Γ) to an arbitrary automorphism T by showing that 2h(T) ^ c(T) for any automorphism.