3 - The Ornstein–Weiss Machinery
Published online by Cambridge University Press: 04 August 2010
Summary
In this section, we describe the constructive tools we will need in order to continue with our work. From the beginnings of the Ornstein approach to constructive ergodic theory and in particular the Isomorphism Theorem it has been understood that there are three basic tools necessary to work constructively with dynamical systems: a version of the Rokhlin lemma; a version of the Ergodic theorem; and a version of the Shannon–McMillan theorem. It has also been understood for some time that a natural context in which all these results hold is that of locally compact and amenable groups. The results described here are lifted almost verbatim from the seminal work of Ornstein and Weiss on this subject. We include them here to provide the reader with ready access to them and as we vary their statements slightly in places. Since we will consider only countable discrete amenable group actions, we do not need the most general form of their results. Thus, for clarity, we have stated these results in the context of discrete group actions. Furthermore, we have opted for a classical description of entropy, using finite partitions and name-counting techniques.
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- Publisher: Cambridge University PressPrint publication year: 2002