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Denjoy systems and dimension groups

Published online by Cambridge University Press:  01 June 2008

KENICHI MASUI ,
FUMIAKI SUGISAKI  and
MASAMICHI YOSHIDA
KENICHI MASUI
Affiliation:
Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 Japan (email: masuikenichi@yahoo.co.jp, yoshida@sci.osaka-cu.ac.jp)
FUMIAKI SUGISAKI
Affiliation:
Department of Mathematics, Kumamoto University, Kurokami, Kumamoto, 860-8555 Japan (email: sugisaki@sci.kumamoto-u.ac.jp)
MASAMICHI YOSHIDA
Affiliation:
Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 Japan (email: masuikenichi@yahoo.co.jp, yoshida@sci.osaka-cu.ac.jp)
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Abstract

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We construct an adic model for the Denjoy system (X,T) of finite double orbit number. It is based on the dual Ostrowski numeration system. By using this construction, we can determine the K-theory of $C(X)\rtimes _T \mathbb {Z}$ associated with (X,T) in an effective way.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 877 - 913
Copyright
Copyright © Cambridge University Press 2008

References

[1]Arnoux, P., Ferenczi, S. and Hubert, P.. Trajectories of rotations. Acta Arith. 87(3) (1999), 209–217.CrossRefGoogle Scholar
[2]Berthé, V., Ferenczi, S. and Zamboni, L. Q.. Interactions between Dynamics, Arithmetics and Combinatorics: the Good, the Bad, and the Ugly (Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005.Google Scholar
[3]Durand, F., Host, B. and Skau, C.. Substitutional dynamical systems, Bratteli diagrams and Dimension groups. Ergod. Th. & Dynam. Sys. 19(4) (1999), 953–993.CrossRefGoogle Scholar
[4]Effros, E. G.. Dimensions and C *-algebras (CBMS Regional Conference Series in Mathematics, 46). Conference Board of the Mathematical Sciences, Washington, D.C., 1981.CrossRefGoogle Scholar
[5]Effros, E. G. and Shen, C.-L.. Approximately finite C *-algebras and continued fractions. Indiana Univ. Math. J. 29 (1980), 191–204.CrossRefGoogle Scholar
[6]Effros, E. G. and Shen, C.-L.. The geometry of finite rank dimension groups. Illinois J. Math. 25(1) (1981), 27–38.CrossRefGoogle Scholar
[7]Fogg, N. P.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Springer, Berlin, 2002.CrossRefGoogle Scholar
[8]Giordano, T., Putnam, I. F. and Skau, C. F.. Topological orbit equivalence and C *-crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
[9]Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3 (1992), 827–864.CrossRefGoogle Scholar
[10]Ito, S.. Some skew product transformation associated with continued fractions and their invariant measures. Tokyo J. Math. 9(1) (1986), 115–133.CrossRefGoogle Scholar
[11]Markley, N. G.. Homeomorphisms of the circle without periodic points. Proc. London Math. Soc. (3) 20 (1970), 688–698.CrossRefGoogle Scholar
[12]Mingo, J. A.. A classification of one-dimensional almost periodic tilings arising from the projection method. Trans. Amer. Math. Soc. 352(11) (2000), 5263–5277.CrossRefGoogle Scholar
[13]Putnam, I., Schmidt, K. and Skau, C.. C *-algebras associated with Denjoy homeomorphisms of the circle. J. Optim. Theory 16 (1986), 99–126.Google Scholar
[14]Vershik, A. M.. A theorem on periodical Markov approximation in ergodic theory. Ergodic Theory and Related Topics (Mathematical Research, 12). Akademie, Berlin, 1982, pp. 195–206.Google Scholar
[15]Vershik, A. M.. Uniform algebraic approximation of shift and multiplication operators. Soviet Math. Dokl. (1) 24 (1981), 97–100.Google Scholar
[16]Vershik, A. M. and Livshits, A. N.. Adic models of ergodic transformations, spectral theory, substitutions, an related topics. Representation Theory and Dynamical Systems (Advances in Soviet Mathematics, 9). American Mathematical Society, Providence, RI, 1992, pp. 185–204.CrossRefGoogle Scholar
[17]Vershik, A. M. and Sidorov, N. A.. Arithmetic expansions associated with a rotation of the circle and with continued fractions. St. Petersburg Math. J. 5 (1994), 1121–1136.Google Scholar