Rotational (and Other) Representations of Stochastic Matrices

Joel E. Cohen (1981) conjectured that any stochastic matrix P = fpi;jg could be represented by some circle rotation f in the following sense: For some partition fSig of the circle into sets consisting of …nite unions of arcs, we have (*) pi;j = (f (Si) \ Sj) = (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, P > 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. Keywords: rotational representation, stochastic matrix, cycle decomposition MSC 2000 subject classi…cations. Primary: 60J10. Secondary: 15A51