Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2005-13

September 2005

Rotational (and Other) Representations of Stochastic Matrices

Steve Alpern and V. S. Prasad

Joel E. Cohen (1981) conjectured that any stochastic matrix P = {pi,j} could be represented by some circle rotation f in the following sense: For some partition {Si} of the circle into sets consisting of finite unions of arcs, we have (*) pi,j = (f (SiSj)/ (Si),, where 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, PN > 0 for some N) stochastic matrix P can be represented (in the sense of * but with Si 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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