On the Rotational Dimension of Stochastic Matrices

Abstract

Let $(S_i, i = 1,2,\ldots, n), n > 1$, be a partition of the circle into sets $S_i$ each consisting of union of $\delta(i) < \infty \operatorname{arcs} A_{kl}$. Let $f_t$ be a rotation of length $t$ of the circle and denote Lebesgue measure by $\lambda$. Then every recurrent stochastic matrix $P$ on $S = \{1,\ldots,n\}$ is given according to a theorem of Cohen $(n = 2)$, Alpern and Kalpazidou $(n \geq 2)$ by $p_{ij} = \lambda(S_i \cap f^{-1}_t(S_j))/\lambda(S_i)$ for some choice of rotation $f_t$ and partition $\mathscr{J} = \{S_i\}$. The number $\delta(\mathscr{J}) = \max_i \delta(i)$ is called the length of description of the partition $\mathscr{J}$. Then it turns out that the minimal value of $\delta(\mathscr{J})$, when $\mathscr{J}$ varies, characterizes the matrix $P$. We call this value the rotational dimension of $P$. We prove that for certain recurrent $n \times n$ stochastic matrices the rotational dimension is provided by the number of Betti circuits of the graph of $P$. One preliminary result shows that there are recurrent $n \times n$ stochastic matrices which admit minimal positive circuit decompositions in terms of the Betti circuits of their graph. Finally, a generalization of the rotational dimension for the transition matrix functions is also given.