Vol. 43, No. 3, 1972

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Entropy and approximation of measure preserving transformations

T. Schwartzbauer

Vol. 43 (1972), No. 3, 753–764

The relationship between the entropy and rate of approximation of an automorphism was first discovered by A. B. Katok. He defined for each automorphism T an invariant e(T) which depends only on the rate of approximation of T and then proved that h(T) c(T) 2h(T) for any ergodic automorphism T, where h(T) denotes the entropy of T. The proof which he gave that e(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(T) = 2h(T) to an arbitrary automorphism T by showing that 2h(T) c(T) for any automorphism.

Mathematical Subject Classification
Primary: 28A65
Received: 19 April 1972
Published: 1 December 1972
T. Schwartzbauer