Recently a new invariant of K-theoretic nature has emerged which is potentially very useful for the study of symbolic systems. We give an outline of the theory behind this invariant. Then we demonstrate the relevance and power of the invariant, focusing on the families of substitution minimal systems and Toeplitz flows.
@article{bwmeta1.element.bwnjournal-article-cmv84i1p203bwm,
author = {Christian Skau},
title = {Ordered K-theoryand minimal symbolic dynamical systems},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {203-227},
zbl = {0972.54027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p203bwm}
}
Skau, Christian. Ordered K-theoryand minimal symbolic dynamical systems. Colloquium Mathematicae, Tome 84/85 (2000) pp. 203-227. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p203bwm/
[000] [1] M. Boyle, Topological orbit equivanlence and factor maps in symbolic dynamics, Ph.D. thesis, Univ. of Washington, 1983.
[001] [2] O. Bratteli, Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195-234. | Zbl 0264.46057
[002] [3] T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 (1991), 241-256. | Zbl 0746.58047
[003] [4] T. Downarowicz and F. Durand, Factors of Toeplitz flows and other almost 1-1 extensions over group rotations, preprint, 1998. | Zbl 1023.37004
[004] [5] T. Downarowicz and Y. Lacroix, Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows, Studia Math. 130 (1998), 149-170. | Zbl 0916.28013
[005] [6] F. Durand, B. Host and C. Skau, Substitution dynamical systems, Bratteli diagrams, and dimension groups, Ergodic Theory Dynam. Systems, to appear. | Zbl 1044.46543
[006] [7] E. G. Effros, Dimensions and C*-algebras, CBMS Regional Conf. Ser. in Math. 46, Conf. Board Math. Sci. and Amer. Math. Soc., 1981.
[007] [8] E. Effros, D. Handelman and C.-L. Shen, Dimension groups and their affine reprensentations, Amer. J. Math. 102 (1980), 385-407. | Zbl 0457.46047
[008] [9] G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. Algebra 38 (1976), 29-44. | Zbl 0323.46063
[009] [10] A. H. Forrest, K-groups associated with substitution minimal systems, Israel J. Math. 98 (1997), 101-139. | Zbl 0891.46042
[010] [11] T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and C*-crossed products, J. Reine Angew. Math. 469 (1995), 51-111. | Zbl 0834.46053
[011] [12] R. Gjerde and O. Johansen, Bratteli-Vershik models for Cantor minimal systems: Applications to Toeplitz flows, NB Ergodic Theory Dynam. Systems, to appear. | Zbl 0992.37008
[012] [13] E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems, Internat. J. Math. 6 (1995), 569-579. | Zbl 0879.54046
[013] [14] D. Handelman, Positive matrices and dimension groups affiliated to topological Markov chains, in: Proc. Sympos. Pure Math. 38, part I, Amer. Math. Soc., 1982, 191-194. | Zbl 0513.46048
[014] [15] T. Harju and M. Linna, On the periodicity of morphisms on free monoids, RAIRO Inform. Théor. Appl. 20 (1986), 47-54. | Zbl 0608.68065
[015] [16] R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups, and topological dynamics, Internat. J. Math. 3 (1992), 827-864. | Zbl 0786.46053
[016] [17] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963.
[017] [18] B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoret. Comput. Sci. 99 (1992), 327-334. | Zbl 0763.68049
[018] [19] --, Reconnaissabilité des substitutions et complexité des suites automatiques, Bull. Soc. Math. France 124 (1996), 101-118.
[019] [20] N. Ormes, Strong orbit realization for minimal homeomorphisms, J. Anal. Math. 71 (1997), 103-133. | Zbl 0881.28013
[020] [21] W. Parry and S. Tuncel, Classification Problems in Ergodic Theory, London Math. Soc. Lecture Note Ser. 67, Cambridge Univ. Press, Cambridge, 1982. | Zbl 0487.28014
[021] [22] I. F. Putnam, K. Schmidt and C. Skau, C*-algebras associated with Denjoy homeomorphisms of the circle, J. Operator Theory 16 (1986), 99-126. | Zbl 0611.46067
[022] [23] M. Queffélec, Substitution Dynamical Systems, Lecture Notes in Math. 1294, Springer, (1987).
[023] [24] F. Sugisaki, The relationship between entropy and strong orbit equivalence for the minimal homeomorphisms (I), Internat. J. Math., to appear. | Zbl 1055.37007
[024] [25] A. M. Vershik, A theorem on the Markov periodical approximation in ergodic theory, J. Soviet Math. 28 (1985), 667-674. | Zbl 0559.47006
[025] [26] P. Walters, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, 1982.
[026] [27] S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete 67 (1984), 95-107. | Zbl 0584.28007