Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T06:40:20.419Z Has data issue: false hasContentIssue false
  • English
  • Français

A Family of Generalized Riesz Products

Published online by Cambridge University Press:  20 November 2018

A. H. Dooley  and
S. J. Eigen
A. H. Dooley
Affiliation:
School of Mathematics, University of New South Wales, Kensington 2033, Australia
S. J. Eigen
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Generalized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

Keywords

28D0342A5547A35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 2 , 01 April 1996 , pp. 302 - 315
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Baxter, J.R.,A class ofergodic automorphisms, Ph.D. thesis, U. of Toronto, Toronto, 1969.Google Scholar
2. Bourgain, J., On the spectral type ofOrnstein s Class one transformations, Israel J. Math. 84 (1993), 53–63.Google Scholar
3. Brown, G. and Dooley, A.H., Odometer actions on G-measures, Ergodic Theory Dynamical Systems 11 (1991), 297–307.Google Scholar
4. Brown, G., Dichotomy theorems for G-measures, Internat. J. Math., to appear.Google Scholar
5. Brown, G. and Moran, W., On orthogonality of Rieszproducts, Math. Proc. Cambridge Philos. Soc. 76 (1974), 173–181.Google Scholar
6. Choksi, J. and Nadkarni, M., The maximal spectral type of a rank one transformation, Canad. Math. Bull. 37 (1994), 29–36.Google Scholar
7. Friedman, N., Mixing and sweeping out, Israel J. Math. 68 (1989), 365–375.Google Scholar
8. Graham, C.C. and McGehee, O.C.,Essays in commutative harmonic analysis, Springer Verlag, 1979.Google Scholar
9. Hewitt, E. and Zuckerman, H., Singular measures with absolutely continuous convolution squares, Math. Proc. Cambridge Philos. Soc. 73 (1973), 307–316.Google Scholar
10. Host, B., Méla, J.F., and Parreau, F., Non-singular transformations and spectral analysis of measures, Bull. Soc. Math. France, 119 (1991), 33–90.Google Scholar
11. Kilmer, S., Equivalence of Riesz products, Contemp. Math. 19 (1989), 101–114.Google Scholar
12. Kilmer, S.J. and Saeki, S., On Riesz product measures, mutual absolute continuity and singularity, Ann. Inst. Fourier (Grenoble) 38-2 (1988), 63–69.Google Scholar
13. Klemes, I. and Reinhold, K., Rank one transformations with singular spectral type, preprint.Google Scholar
14. Omstein, D., On the root problem in ergodic theory, Proc. Sixth Berkeley Symp. Math. Stat, and Prob. Vol II, 347–356.Google Scholar
15. Parreau, F., Ergodicite etpurete desproduits de Riesz, Ann. Inst. Fourier (Grenoble) 40-2 (1990), 391–405.Google Scholar
16. Peyriere, J., Sur lesproduits de Riesz, C. R. Acad. Sci. Paris Ser. A-B 276 (1973), 1453–1455.Google Scholar
17. Peyriere, J., Etude de quelquesproprietes desproduits de Riesz, Ann. Inst. Fourier (Glenoble) 25-2 (1975), 127–169.Google Scholar
18. Ritter, G., On dichotomy of Riesz products, Math. Proc. Cambridge Philos. Soc. 85 (1978), 79–90.Google Scholar
19. Zygmund, A., On lacunary trignometric series, Trans. Amer. Math. Soc. 34 (1932), 435–446.Google Scholar
20. Zygmund, A.,Trigonometric Series, Cambridge University Press, 1968.Google Scholar