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A note on roots of Markov shifts

Published online by Cambridge University Press:  14 July 2016

S. M. Rudolfer
S. M. Rudolfer*
Affiliation:
University of Manchester
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Abstract

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.

Keywords

ROOTS OF TRANSFORMATIONSMARKOV SHIFTSBERNOULLI SHIFTSIMBEDDING TRANSFORMATIONS IN FLOWSERGODIC AUTOMORPHISMS WITH DISCRETE SPECTRUMK-AUTOMORPHISMSENTROPYCOMPLETE METRIC INVARIANTSTOCHASTIC PROCESSCARTESIAN PRODUCTROTATION ON A CYCLIC GROUP
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 591 - 596
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Adler, R. L., Shields, P. and Smorodinsky, M. (1972) Irreducible Markov shifts. Ann. Math. Statist. 43, 1027–1029.Google Scholar
[2] Billingsley, P. (1965) Ergodic Theory and Information. Wiley, New York.Google Scholar
[3] Blum, J. R. and Friedman, N. A. (1966) On commuting transformations and roots. Proc. Amer. Math. Soc. 17, 1370–1374.Google Scholar
[4] Feller, W. (1968) Introduction to Probability Theory and its Applications. Vol. I, 3rd ed. Wiley, New York.Google Scholar
[5] Halmos, P. R. (1942) Square roots of measure-preserving transformations. Amer. J. Math. 64, 153–166.CrossRefGoogle Scholar
[6] Halmos, P. R. (1956) Lectures on ergodic theory. Mathematical Society of Japan, Tokyo.Google Scholar
[7] Krengel, U. and Michel, H. (1967) Uber Wurzeln ergodischer Transformationen. Math. Zeitschrift, 96, 50–57.Google Scholar
[8] Ornstein, D. S. (1974) Ergodic Theory, Randomness and Dynamical Systems. Yale University Press, New Haven.Google Scholar
[9] Ornstein, D. S. (1973) A K-automorphism with no square root and Pinsker's conjecture. Adv. Math. 10, 89–102.Google Scholar
[10] Smorodinsky, M. (1970) Ergodic Theory and Entropy. Lecture Notes in Mathematics 214, Springer, Berlin.Google Scholar