Abstract
Two ratio limit concepts for transformations preserving infinite measures, rational ergodicity and bounded rational ergodicity, are discussed and compared. The concept of rational ergodicity is used to construct some continuous measures on the circle, which show that the exceptional set in the weak mixing theorem may be rather large.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Aaronson,Rational ergodicity and a metric invariant for Markov shifts, Israel J. Math.27 (1977), 93–123.
J. Aaronson,On the ergodic theory of non-integrable functions and infinite measure spaces, Israel J. Math.27 (1977), 163–173.
J. Aaronson,Sur le jeu de Saint-Petersbourg, C. R. Acad. Sci. ParisA286 (1978), 839–842.
A. Beck,Eigenoperators of ergodic transformations, Trans. Amer. Math. Soc.94 (1960), 118–129.
S. Foguel and M. Lin,Some ratio limit theorems for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete23 (1972), 55–66.
H. Furstenberg and B. Weiss,The finite multipliers of infinite ergodic transformations, to appear.
A. Hajian and S. Kakutani,An example of an ergodic measure preserving transformation on a infinite measure space, Springer Lecture Notes160, 1970, pp. 45–52.
S. Kakutani,Induced measure preserving transformations, Proc. Imp. Acad. Sci. Tokyo19 (1943), 635–641.
E. Seneta,Regularly varying functions, Springer Lecture Notes508, 1976.
J. G. Sinai,Theory of Dynamical Systems, Part I, Aarhus University Press, 1970.
W. Feller,An Introduction to Probability Theory and its Applications, Vol. II, Wiley, New York, 1971.
Author information
Authors and Affiliations
Additional information
To the memory of Shlomo Horowitz
Rights and permissions
About this article
Cite this article
Aaronson, J. Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle. Israel J. Math. 33, 181–197 (1979). https://doi.org/10.1007/BF02762160
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02762160