Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-08T03:44:59.620Z Has data issue: false hasContentIssue false
  • English
  • Français

Anzai and Furstenberg Transformations on the 2-Torus and Topologically Quasi-Discrete Spectrum

Published online by Cambridge University Press:  20 November 2018

Kazunori Kodaka
Kazunori Kodaka*
Affiliation:
Department of Mathematics, College of Science Ryukyu University, Nishihara-cho, Okinawa 903-01, Japan
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.

Let ϕ0 be an Anzai transformation on the 2-torus T2 defined by ϕ0(x,y) = (e2πiθx,xy) and ϕy a Furstenberg transformation on T2 defined by ϕf(x,y) = (e2πiθx,e2πif(x)xy) where θ is an irrational number and f is a real valued continuous function on the 1-torus T. In the present note we will show that ϕf has topologically quasi-discrete spectrum if and only if ϕf is topologically conjugate to ϕ0. Furthermore we will show that for any irrational number θ there is a real valued continuous function f on T such that ϕf does not have topologically quasi-discrete spectrum but is uniquely ergodic.

Keywords

46L8046L40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 1 , 01 March 1995 , pp. 87 - 92
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Baggett, L. and Merrill, K., On the cohomological equivalence of a class of functions under an irrational rotation of bounded type, Proc. Amer. Math. Soc. 111(1991), 787–793.Google Scholar
2. Effros, E. G. and Hahn, F., Locally compact transformation groups and C*-algebras, Mem. Amer. Math. Soc. 75(1967).Google Scholar
3. Furstenberg, H., Strict ergodicity and transformation of the torus, Amer. J. Math. 83(1961), 573–601.Google Scholar
4. Gottschalk, W. H. and Hedlund, G. A., Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 36(1955).Google Scholar
5. Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, Oxford at the Clarendon Press, 1979.Google Scholar
6. Kodaka, K., Diffeomorphisms of irrational roation C* -algebras by non-generic rotations II, J. Operator Theory, to appear.Google Scholar
7. Lang, S., Introduction to Diophantine Approximations, Addison-Wesley, 1966.Google Scholar
8. Parry, W., Topics in Ergodic Theory, Cambridge University Press, 1981.Google Scholar
9. Pedersen, G. K., C*-Algebras and their Automorphism Groups, Academic Press, 1979.Google Scholar
10. Rouhani, H., A Furstenberg transformation of the 2-torus without quasi-discrete spectrum, Canad. Math. Bull. 33(1990), 316–322.Google Scholar
11 Rouhani, H., Quasi-roation C*-algebras, Pacific J. Math. 148(1991), 131–151.Google Scholar