## Rotational representations of stochastic matrices.(English)Zbl 0573.15013

Let $$P=(p_{ij})$$ be an $$n\times n$$ stochastic matrix with a positive stationary row eigenvector $$\pi =(\pi_ 1,\pi_ 2,...,\pi_ n)$$. Let J be a partition of the unit circle into sets $$J_ 1,J_ 2,...,J_ n$$ with $$m(J_ i)=\pi_ i$$, where m is the Lebesgue measure. Let $$f_ t(x)$$ indicate a rotation through distance t. It has been shown that for any such P each $$p_{ij}$$ has the form $$m(f_ t(J_ i)\cap J_ j)/m(J_ i)$$ where each $$J_ i$$ is the union of a bounded number of arcs. It had been conjectured that if this bound were b(n), then $$b(n)=n- 1$$. The conjecture was later shown to be false for $$n=6$$ and $$n\geq 8$$. It had been also shown that $$b(2)=1$$. In this paper it is shown that $$b(3)=2$$, and that for any n, b(n)$$\geq b(n-1)$$. The paper includes the following conjecture: let P be a reducible stochastic matrix corresponding to a reducible Markov chain with cyclic classes of sizes $$1,c_ 1,c_ 2,...,c_ r$$. Then if $$p=c_ 1+c_ 2+...+c_ r$$ and $$H(p)=Max\{l.c.m.(c_ 1,c_ 2,...,c_ r):$$ all partitions$$\}$$, it is conjectured that $$b(n)=H(n-1)$$.
Reviewer: R.Sinkhorn

### MSC:

 15B51 Stochastic matrices 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 28D05 Measure-preserving transformations
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