Rotational Representations of Stochastic Matrices

Abstract

Let $(p_{ij})$ be the matrix of a recurrent Markov chain with stationary vector $\pi > 0,$ and let $\{J_i\}$ be a partition of the unit circle into sets $J_1, \cdots, J_n, m(J_i) = \pi_i$, where $m$ is Lebesgue measure. Suppose $f_t$ defines rotation through distance $t$. The conditions under which $p_{ij}$ can be written as $m(f_t(J_i) \cap J_j)/m(J_i)$ for all $i$ and $j$, where each $J_i$ is the union of at most $b(n)$ arcs, have recently been examined by Steve Alpern and Joel Cohen. Cohen conjectured that $b(n) = n - 1$, and proved $b(2) = 1$. Alpern proved that Cohen's conjecture was false for $n$ sufficiently large, and gave bounds for $b(n)$. We give a construction that shows that $b(3) = 2$, and prove that $b(n)$ is nondecreasing.