Abstract
The eigenvalues of a non-singular conservative ergodic transformation of a separable measure space form a Borel subgroup of the circle of measure zero. We show that this is the only metric restriction on their size. However, the larger the eigenvalue group of the transformation, the “less recurrent” it is.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Aaronson,Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, Isr. J. Math.33 (1979), 181–197.
W. Feller,An Introduction to Probability Theory and its Applications, Vol. II, J. Wiley, New York, 1966, 1971.
O. Frostman,Potentiel d’equilibre et capacite des ensembles, Thesis, Lund. 1935.
H. Furstenberg and B. Weiss,The finite multipliers of infinite ergodic transformations, Lecture Notes668, Springer, Berlin, 1978, pp. 127–132.
M. Kac,On the notion of recurrence in discrete stochastic processes, Bull. Am. Math. Soc.53 (1947), 1002–1010.
J. P. Kahane and R. Salem,Ensembles parfaites et series trigonometriques, Hermann, Paris, 1963.
S. Kakutani,Induced measure preserving transformations, Proc. Imp. Acad. Sci. Tokyo19 (1943), 635–641.
U. Krengel,Classification of states for operators, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1966, pp. 415–428.
K. Schmidt,Spectra of ergodic group actions, Isr. J. Math.41 (1982), 151–153.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aaronson, J. The eigenvalues of non-singular transformations. Israel J. Math. 45, 297–312 (1983). https://doi.org/10.1007/BF02804014
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02804014