Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200513September 2005 
Rotational (and Other) Representations of Stochastic Matrices
Steve Alpern and V. S. Prasad
Joel E. Cohen (1981) conjectured that any stochastic matrix P = {p_{i,j}} could be represented by some circle rotation f in the following sense: For some partition {S_{i}} of the circle into sets consisting of finite unions of arcs, we have (*) p_{i,j} = µ (f (S_{i}) S_{j})/µ (S_{i}),, where µ denotes arc length. In this paper we show how cycle decomposition techniques originally used (Alpern, 1983) to establish Cohen's conjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, P^{N} > 0 for some N) stochastic matrix P can be represented (in the sense of * but with S_{i }merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue probability space. Representations by pointwise and setwise periodic automorphisms are also established. While this paper is largely expository, all the proofs, and some of the results, are new.
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Last modified: 31 October 2005