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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
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Abstract

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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

Information

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

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