## 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:

  Akcoglu, M. A., del Junco, A. and Rahe, M. (1979). Finitary codes between Markov processes. Z. Wahrsch. Verw. Gebiete 47 305-314. · Zbl 0403.28017  Angel, O., Holroyd, A. E. and Soo, T. (2011). Deterministic thinning of finite Poisson processes. Proc. Amer. Math. Soc. 139 707-720. · Zbl 1208.60047  Ball, K. (2005). Monotone factors of i.i.d. processes. Israel J. Math. 150 205-227. · Zbl 1146.37010  Ball, K. (2005). Poisson thinning by monotone factors. Electron. Commun. Probab. 10 60-69 (electronic). · Zbl 1110.60050  Blum, J. R. and Hanson, D. L. (1963). On the isomorphism problem for Bernoulli schemes. Bull. Amer. Math. Soc. 69 221-223. · Zbl 0121.13601  Breiman, L. (1957). The individual ergodic theorem of information theory. Ann. Math. Statist. 28 809-811. · Zbl 0078.31801  Burton, R. and Rothstein, A. (1977). Isomorphism theorems in ergodic theory. Technical report, Oregon State Univ., Corvallis, OR.  Burton, R. M., Keane, M. S. and Serafin, J. (2000). Residuality of dynamical morphisms. Colloq. Math. 84/85 307-317. · Zbl 0963.37007  Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Ya. G. (1982). Ergodic Theory. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 245 . Springer, New York. Translated from the Russian by A. B. Sosinskiĭ. · Zbl 0493.28007  del Junco, A. (1981). Finitary codes between one-sided Bernoulli shifts. Ergodic Theory Dynam. Systems 1 285-301. · Zbl 0486.28015  del Junco, A. (1990). Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic. Ergodic Theory Dynam. Systems 10 687-715. · Zbl 0697.58032  Denker, M. and Keane, M. (1979). Almost topological dynamical systems. Israel J. Math. 34 139-160. · Zbl 0441.28008  de la Rue, T. (2006). An introduction to joinings in ergodic theory. Discrete Contin. Dyn. Syst. 15 121-142. · Zbl 1105.37003  Downarowicz, T. (2011). Entropy in Dynamical Systems. New Mathematical Monographs 18 . Cambridge Univ. Press, Cambridge. · Zbl 1220.37001  Dudley, R. (1989). Real Analysis and Probability . Wadsworth, Pacific Grove, CA. · Zbl 0686.60001  Evans, S. N. (2010). A zero-one law for linear transformations of Lévy noise. In Algebraic Methods in Statistics and Probability II. Contemp. Math. 516 189-197. Amer. Math. Soc., Providence, RI. · Zbl 1206.60037  Gurel-Gurevich, O. and Peled, R. (2013). Poisson thickening. Israel J. Math. 196 215-234. · Zbl 1306.60052  Hall, P. (1935). On representatives of subsets. J. London Math. Soc. (1) 10 26-30. · Zbl 0010.34503  Halmos, P. R. (1961). Recent progress in ergodic theory. Bull. Amer. Math. Soc. 67 70-80. · Zbl 0161.11401  Harvey, N., Holroyd, A. E., Peres, Y. and Romik, D. (2007). Universal finitary codes with exponential tails. Proc. Lond. Math. Soc. (3) 94 475-496. · Zbl 1148.37004  Holroyd, A. E., Lyons, R. and Soo, T. (2011). Poisson splitting by factors. Ann. Probab. 39 1938-1982. · Zbl 1277.60087  Katok, A. (2007). Fifty years of entropy in dynamics: 1958-2007. J. Mod. Dyn. 1 545-596. · Zbl 1149.37001  Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352-371. · Zbl 0357.94012  Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 397-406. · Zbl 0405.28017  Krieger, W. (1970). On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149 453-464. · Zbl 0204.07904  Mešalkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk SSSR 128 41-44. · Zbl 0099.12301  Mester, P. (2013). Invariant monotone coupling need not exist. Ann. Probab. 41 1180-1190. · Zbl 1300.05276  Meyerovitch, T. (2013). Ergodicity of Poisson products and applications. Ann. Probab. 41 3181-3200. · Zbl 1279.60061  Ornstein, D. (1970). Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 337-352. · Zbl 0197.33502  Ornstein, D. (2013). Newton’s laws and coin tossing. Notices Amer. Math. Soc. 60 450-459. · Zbl 1358.37007  Ornstein, D. S. (1974). Ergodic Theory , Randomness , and Dynamical Systems. Yale Mathematical Monographs 5 . Yale Univ. Press, New Haven, CT. · Zbl 0296.28016  Ornstein, D. S. and Weiss, B. (1975). Unilateral codings of Bernoulli systems. Israel J. Math. 21 159-166. · Zbl 0323.28008  Ornstein, D. S. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1-141. · Zbl 0637.28015  Petersen, K. (1989). Ergodic Theory. Cambridge Studies in Advanced Mathematics 2 . Cambridge Univ. Press, Cambridge. Corrected reprint of the 1983 original. · Zbl 0676.28008  Propp, J. G. (1991). Coding Markov chains from the past. Israel J. Math. 75 289-328. · Zbl 0766.28006  Quas, A. and Soo, T. (2014). Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc. To appear. Available at . · Zbl 1354.37010  Rudolph, D. J. (1981). A characterization of those processes finitarily isomorphic to a Bernoulli shift. In Ergodic Theory and Dynamical Systems , I ( College Park , Md. , 1979 - 1980). Progr. Math. 10 1-64. Birkhäuser, Boston, MA. · Zbl 0483.28015  Rudolph, D. J. (1990). Fundamentals of Measurable Dynamics : Ergodic Theory on Lebesgue Spaces . Clarendon, New York. · Zbl 0718.28008  Serafin, J. (2006). Finitary codes, a short survey. In Dynamics & Stochastics. Institute of Mathematical Statistics Lecture Notes-Monograph Series 48 262-273. IMS, Beachwood, OH. · Zbl 1132.37005  Shea, S. (2013). Finitarily Bernoulli factors are dense. Fund. Math. 223 49-54. · Zbl 1347.37014  Shea, S. M. (2012). On the marker method for constructing finitary isomorphisms. Rocky Mountain J. Math. 42 293-304. · Zbl 1246.37016  Sinaĭ, Ja. G. (1964). On a weak isomorphism of transformations with invariant measure. Mat. Sb. ( N.S. ) 63 (105) 23-42.  Sinai, Y. G. (2010). Selecta. Volume I. Ergodic Theory and Dynamical Systems . Springer, New York. · Zbl 1201.01054  Srivastava, S. M. (1998). A Course on Borel Sets. Graduate Texts in Mathematics 180 . Springer, New York. · Zbl 0903.28001  Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423-439. · Zbl 0135.18701  Thorisson, H. (2000). Coupling , Stationarity , and Regeneration . Springer, New York. · Zbl 0949.60007  Weiss, B. (1972). The isomorphism problem in ergodic theory. Bull. Amer. Math. Soc. 78 668-684. · Zbl 0255.28014
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