Ergodicity of the Cartesian product

Flytzanis, Elias G.

doi:10.1090/S0002-9947-1973-0328021-6

Ergodicity of the Cartesian product

Author: Elias G. Flytzanis
Journal: Trans. Amer. Math. Soc. 186 (1973), 171-176
MSC: Primary 28A65
DOI: https://doi.org/10.1090/S0002-9947-1973-0328021-6
MathSciNet review: 0328021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: ${h_1}$ is an ergodic conservative transformation on a $\sigma$ -finite measure space and ${h_2}$ is an ergodic measure preserving transformation on a finite measure space. We study the point spectrum properties of ${h_1} \times {h_2}$ . In particular we show ${h_1} \times {h_2}$ is ergodic if and only if ${h_1} \times {h_2}$ have no eigenvalues in common other than the eigenvalue 1. The conditions on ${h_1},{h_2}$ stated above are in a sense the most general for the validity of this result.

[1] Anatole Beck, Eigen operators of ergodic transformations, Trans. Amer. Math. Soc. 94 (1960), 118–129. MR 0152631, https://doi.org/10.1090/S0002-9947-1960-0152631-3
[2] R. V. Chacon, A class of linear transformations, Proc. Amer. Math. Soc. 15 (1964), 560–564. MR 0165071, https://doi.org/10.1090/S0002-9939-1964-0165071-7
[3] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
[4] Nathaniel A. Friedman, Introduction to ergodic theory, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. Van Nostrand Reinhold Mathematical Studies, No. 29. MR 0435350
[5] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 0213508, https://doi.org/10.1007/BF01692494
[6] Paul R. Halmos, Lectures on ergodic theory, Chelsea Publishing Co., New York, 1960. MR 0111817
[7] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
[8] K. Jacobs, Lecture notes on ergodic theory, 1962/63. Parts I, II, Matematisk Institut, Aarhus Universitet, Aarhus, 1963. MR 0159922
[9] Branko Kronfeld, Ergodic transformations and independence, Glasnik Mat. Ser. III 5 (25) (1970), 221–226 (English, with Serbo-Croatian summary). MR 0276442
[10] S. Kakutani and W. Parry, Infinite measure preserving transformations with “mixing”, Bull. Amer. Math. Soc. 69 (1963), 752–756. MR 0153815, https://doi.org/10.1090/S0002-9904-1963-11022-8
[11] Donald S. Ornstein, On invariant measures, Bull. Amer. Math. Soc. 66 (1960), 297–300. MR 0146350, https://doi.org/10.1090/S0002-9904-1960-10478-8
[12] Frank Hahn and William Parry, Some characteristic properties of dynamical systems with quasi-discrete spectra, Math. Systems Theory 2 (1968), 179–190. MR 0230877, https://doi.org/10.1007/BF01692514