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On Finite Coding Factors of a Class of Random Markov Chains

Published online by Cambridge University Press:  20 November 2018

M. Rahe
M. Rahe*
Affiliation:
Department of Mathematics, Texas A&M University College Station, Texas U.S.A.
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Abstract

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For k-step Markov chains, factors generated by finite length codes split off with Bernoulli complement when maximal in entropy. Those not maximal are relatively finite in another factor which generates or splits off.

These results extend to random Markov chains with finite expected step size, implying that random Markov chains with finite expected step size can have only finitely many ergodic components, each of which is isomorphic to a finite rotation, a Bernoulli shift, or a direct product of a Bernoulli shift with a finite rotation. This result limits the type of zero entropy factors which occur in random Markov chains with finite expected step size, providing a counterpoint to the work of Kalikow, Katznelson, and Weiss, who have shown that each zero entropy process can be embedded in some random Markov chain.

Extending Rudolph and Schwarz, random Markov chains with finite expected step size are limits in of their canonical Markov approximants. The -closure of the class is the Bernoulli cross Generalized Von Neuman processes.

Finitary isomorphism of aperiodic ergodic random Markov chains with finite expected step size is considered.

Applications are made to a class of generalized baker's transformations.

Keywords

28A65generalized baker's transformationrandom Markov chainsuniform martingalefinite coding factorsd̄-limitfinitarily isomorphic
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 3 , 01 September 1994 , pp. 399 - 407
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Adler, R. L., Shields, P. and Smorodinsky, M., Irreducible Markov shifts, Annals of Math. Statist. 43(1972), 1027–1029.Google Scholar
2. Bose, C. J., Generalized bakers transformations, Ergodic Theory Dynamical Systems 9(1989), 1–17.Google Scholar
3. Junco, Andres del, Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic, Ergodic Theory and Dynamical Systems 10(1990), 687–715.Google Scholar
4. del, A. Junco and Rahe, M., Finitary codings and weak Bernoulli partitions, Proc. Amer. Math. Soc. 75 (1979), 259–264.Google Scholar
5. Kalikow, S., Random Markov processes and uniform martingales, Israel J. Math. 71(1990), 33–54.Google Scholar
6. Kalikow, S., Katznelson, Y. and Weiss, B., Finitarily deterministic generators for zero entropy systems, Israel J. Math. 79(1992), 33–45.Google Scholar
7. Rahe, M., Relatively finitely determined implies relatively very weak Bernoulli, Canad. J. Math. 30(1978), 531–548.Google Scholar
8. Rahe, M., Finite coding factors of Markov generators, Israel J. Math. 32(1979), 349–355.Google Scholar
9. Rahe, M., On a class of generalized baker's transformations, Canad. J. Math. 45(1993), 638–649.Google Scholar
10. Rudolph, Daniel J., A characterization of those processes finitarily isomorphic to a Bernoulli shift, Ergodic Theory Dynamical Systems 1(1979), 1–64.Google Scholar
11. Rudolph, Daniel J. and Schwarz, Gideon, The limits in of multi-step Markov chains, Israel J. Math. 28(1977), 103–109.Google Scholar
12. Thouvenot, J.-R, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l'un est un schéma de Bernoulli, Israel J. Math. 21(1975), 177–207.Google Scholar