A monotone Sinai theorem.(English)Zbl 1359.37014

Let $$p=(p_i)_{i\in [N]}$$ be any probability vector on the state set $$[N]:= \{0,1,\dots,N-1\}$$. For each $$n\geq 0$$, define $$\mu_p(_m[i_0,\dots,i_n])=p_{i_0}p_{i_1}\dots p_{i_n}$$, where $$i_0,i_1,\dots,i_n \in [N]$$. Such a measure $$\mu_p$$ is called a Bernoulli measure. Then $$B(p):= ([N]^{\mathbb Z},\mu_p,S)$$ is called a Bernoulli shift on $$N$$ symbols, where $$S$$ is the left-shift given by $$S(y)_i=y_{i+1}$$. The entropy of the Bernoulli shift $$B(p)$$ is given by the positive number $$H(p):=-\sum^{n-1}_{i=0}p_i\log p_i$$. Let $$(X,\mu)$$ be a probability space. If $$T:X\rightarrow X$$ is a map such that $$\mu\circ T^{-1}=\mu$$, then $$(X,\mu,T)$$ is a measure-preserving system. Ya. G. Sinai [Sov. Math., Dokl. 3, 1725–1729 (1963; Zbl 0205.13501); translation from Dokl. Akad. Nauk SSSR 147, 797–800 (1962)] has proved that if $$(X,\mu,T)$$ is a nonatomic invertible ergodic measure-preserving system of entropy $$h>0$$, then it has any Bernoulli shift of any entropy $$h' \leq h$$ as a factor. If $$p$$ and $$q$$ are two probability measures on $$[N]:= \{0, 1,\dots,N - 1\}$$ such that $$p$$ stochastically dominates $$q$$ ($$\sum^k_{i=0} p_i\leq \sum^k_{i=0}q_i$$ for all $$0\leq k < N$$) and $$H(p) > H(q)$$, does there exist a monotone factor map from $$B(p)$$ to $$B(q)$$? In the present paper, the authors answer above question affirmatively. One of the main results in the paper is following:
Theorem 1. Let $$B(p)$$ and $$B(q)$$ be Bernoulli shifts with symbols in $$[N]$$ (where one allows the possibility that $$p$$ and $$q$$ give zero mass to some symbols). If the entropy of $$B(p)$$ is strictly greater than that of $$B(q)$$ and the measure $$p$$ stochastically dominates $$q$$, then $$B(q)$$ is a monotone factor of $$B(p)$$.
Secondly, the authors prove the following related result in the more restricted context of Bernoulli factors.
Theorem 2. Let $$B(p)$$ and $$B(q)$$ be Bernoulli shifts with symbols in $$[N]$$ (where one allows the possibility that $$p$$ and $$q$$ give zero mass to some symbols). Let $$R$$ be any relation on $$[N]$$. If the entropy of $$B(p)$$ is strictly greater than that of $$B(q)$$, and the measure $$p$$ $$R$$-dominates the measure $$q$$, then there exists a factor $$\phi$$ from $$B(p)$$ to $$B(q)$$ such that $$(x_0,\phi(x_0))\in R$$ for all $$x\in [N]^{\mathbb{Z}}$$.
The paper is well written, the results are interesting in the sense that they suggest the existence of factor $$\phi$$ from $$B(p)$$ to $$B(q)$$ satisfying the conditions of the theorems.

MSC:

 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 60G10 Stationary stochastic processes 60E15 Inequalities; stochastic orderings

Zbl 0205.13501
Full Text:

References:

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