Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T13:26:53.584Z Has data issue: false hasContentIssue false

An example of an automorphism which is a mixing of all degrees but which is not a Kolmogorov automorphism

Published online by Cambridge University Press:  24 October 2008

J. K. Dugdale
Affiliation:
Imperial College, London

Extract

Rohlin (7) has shown that Kolmogorov automorphisms are mixings of all degrees. Later Sucheston(9) introduced regular automorphisms and after showing that they are mixings of all degrees asks if the converse of either of the above theorems holds. Since the equivalence of Kolmogorov and regular automorphisms follows from (1) and (8) we have only one problem. We show that the converse of either of the above theorems is false by constructing a stationary Gaussian process which is a mixing of all degrees but which is not a Kolmogorov automorphism.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1966

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Blum, J. R. and Hanson, D. L.Some asymptotic convergence theorems. Proceedings of an International Symposiumheld at Tulane University,New Orleans (1961).Google Scholar
(2)Doob, J. L.Stochastic processes (New York, 1953).Google Scholar
(3)Fomin, S. V.Dynamical systems in a function space. Ukrain Mat. Ž. 2, no. 2 (1950), 2547.Google Scholar
(4)Jacobs, K. Lecture notes on Ergodic Theory. Matematisk Institut, Aarhus Universitet (1962/1963).Google Scholar
(5)Kolmogorov, A. N.A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces. Dokl. Akad. Nauk SSSR 119 (1958), 861864.Google Scholar
(6)Leonov, V. P.The use of the characteristic functional and semi-invariants in the ergodic theory of stationary processes. Dokl. Akad. Nauk SSSR 133 (1960), 523526.Google Scholar
(7)Rohlin, V. A.Exact endomorphisms of a Lebesgue space. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499530.Google Scholar
(8)Sinai, Ja. G.Probabilistic ideas in Ergodic Theory. Proceedings of the International Congress of Mathematicians (1962).Google Scholar
(9)Sucheston, L.Remarks on Kolmogorov automorphisms. Proceedings of an International Symposiumheld at Tulane University, New Orleans (1961).Google Scholar
(10)Zygmund, A.Trigonometric series (Cambridge, 1959).Google Scholar