Rotational Representations of Stochastic Matrices

Abstract

Let $\{S_i\}, i = 1, \cdots, n$, be a partition of the circle into sets $S_i$ each consisting of a finite union of arcs. Let $f$ be a rotation of the circle and let $u$ denote Lebesgue measure. Then the matrix $P$ defined by $p_{ij} = u(S_i \cap f^{-1} S_j)/u(S_i)$ is stochastic. We prove (and improve) a conjecture of Joel E. Cohen asserting that every irreducible stochastic matrix arises from a construction of this type.