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MultiTowers, conjugacies and codes: Three theorems in ergodic theory, one variation on Rokhlin's Lemma

Authors: S. Alpern and V. S. Prasad
Journal: Proc. Amer. Math. Soc. 136 (2008), 4373-4383
MSC (2000): Primary 37A05; Secondary 60J10
Published electronically: July 8, 2008
MathSciNet review: 2431052
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Abstract: We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation $ \tau$ of a Lebesgue probability space $ (X, \mathcal{A}, \mu) $ are equivalent. One theorem asserts that the conjugates of $ \tau$ are dense in the uniform topology on the space of automorphisms. The other two results assert the existence of a partition of the space $ X$ with special properties. One partition result shows that given a mixing Markov chain, there is a code (i.e., a partition of the space) so that the itinerary process given by $ \tau$ and the partition has the distribution of the given Markov Chain. The other partition result is a generalization of the Rokhlin Lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance. Thus the first two results are equivalent to this strengthening of Rokhlin's Lemma.

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Additional Information

S. Alpern
Affiliation: Delft Institute of Applied Mathematics, P. O. Box 5031, 2600 GA Delft, Netherlands
Address at time of publication: London School of Economics, London WC2A 2AE, United Kingdom

V. S. Prasad
Affiliation: Department of Mathematics, University of Massachusetts Lowell, One University Avenue, Lowell, Massachusetts 01854

Keywords: Rokhlin towers, conjugacy, coding, stationary aperiodic irreducible Markov chain
Received by editor(s): November 13, 2007
Published electronically: July 8, 2008
Dedicated: We dedicate this paper to the memory of Shizuo Kakutani. We miss his kind manner, gentle presence and keen insight.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.