Entropy and approximation of measure preserving transformations

T. Schwartzbauer

Vol. 43 (1972), No. 3, 753–764

DOI: 10.2140/pjm.1972.43.753

Abstract

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

Milestones

Received: 19 April 1972

Published: 1 December 1972