Abstract
We study self-homeomorphisms of zero dimensional metrizable compact Hausdorff spaces by means of the ordered first cohomology group, particularly in the light of the recent work of Giordano Putnam, and Skau on minimal homeomorphisms. We show that flow equivalence of systems is analogous to Morita equivalence between algebras, and this is reflected in the ordered cohomology group. We show that the ordered cohomology group is a complete invariant for flow equivalence between irreducible shifts of finite type; it follows that orbit equivalence implies flow equivalence for this class of systems. The cohomology group is the (pre-ordered) Grothendieck group of the C*-algebra crossed product, and we can decide when the pre-ordering is an ordering, in terms of dynamical properties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
[AR] P. Arnoux and G. Rauzy,Représentation géometrique des suites de complexité 2n+1, Bulletin de la Société Mathématique de France119 (1991), 199–215.
[Bla] B. Blackadar,K-Theory for Operator Algebras, MSRI Publications, Vol. 5, 1986.
[Bw] R. Bowen,Equilibrium States and Ergodic Theory of Anasov Diffeormorphisms, Springer Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975.
[BF] R. Bowen and J. Franks,Homology for zero-dimensional basic sets Annals of Mathematics106 (1977), 73–92.
[B1] M. Boyle,Topological orbit equivalence and factor maps in symbolic dynamics, Ph.D. Thesis, University of Washington, 1983.
[B2] M. Boyle,Symbolic dynamics and matrices, Proceedings of a Conference on Linear Algebra, Graph Theory and Related Topics (R. Brualdi, S. Friedland and V. Klee, eds.), I.M.A. volumes in Mathematics, Minneapolis, Vol. 50, 1993, pp. 1–38.
[BH1] M. Boyle and D. Handelman,Algebraic shift equivalence and primitive matrices, Transactions of the American Mathematical Society336 (1993), 121–149.
[BH2] M. Boyle and D. Handelman,Entropy versus orbit equivalence for minimal homeomorphisms, Pacific Journal of Mathematics164 (1994), 1–13.
[C] C. Conley,Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, Vol. 38, AMS, Providence, 1978.
[Co] A. Connes,An analogue of the Thom isomorphism for crossed products of a C *-algebra by an action ofR, Advances in Mathematics39 (1981), 31–55.
[DGS] M. Denker, C. Grillenberger and K. Sigmund,Ergodic Theory on Compact Spaces, Springer Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin, 1976.
[EHS] E. Effros, D. Handelman and C.-L. Shen,Dimension groups and their affine representations, American Journal of Mathematics102 (1980), 385–407.
[F1] J. Franks,Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, Vol. 49, AMS, Providence, 1982.
[F2] J. Franks,Flow equivalence of subshifts of finite type, Ergodic Theory and Dynamical Systems4 (1984), 53–66.
[Fr] D. Fried,The geometry of cross sections to flows, Topology21 (1982), 353–371.
[GPS] T. Giordano, I. Putnam and C. Skau,Topological orbit equivalence and C *-crossed products, to appear in Crelle's Journal.
[G] K. R. Goodearl,Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, Vol. 20, AMS Publications, 1986.
[GH] K. R. Goodearl and D. E. Handelman,Stenosis in dimension groups and AF C *-algebras, Crelle's Journal332 (1982), 1–98.
[GH1] K. R. Goodearl and D. E. Handelman,Rank functions and K 0 of regular rings, Journal of Pure and Applied Algebra7 (1976), 195–216.
[Gr] P. A. Griffith,Infinite Abelian Group Theory, The University of Chicago Press, 1970.
[HPS] R. Herman, I. Putnam and C. Skau,Ordered Bratteli diagrams, dimension groups and topological dynamics, International Journal of Mathematics3 (1992), 827–864.
[LM] D. Lind and B. Marcus,An Introduction to Symbolic Dynamics, Cambridge University Press, to appear.
[M] B. Marcus,Factors and extensions of full shifts, Monatshefte für Mathematik88 (1979), 239–247.
[PT] W. Parry and S. Tuncel,Classification Problems in Ergodic Theory, London Mathematical Society Lecture Notes, Vol. 67, Cambridge University Press, 1982.
[Pi] M. Pimsner,Embedding some transformation group C *-algebras into AF-algebras, Ergodic Theory and Dynamical Systems3 (1983), 113–126.
[Po] Y. T. Poon,A K-theoretic invariant for dynamical systems, Transactions of the American Mathematical Society311 (1989), 515–533.
[Pu] I. Putnam,The C *-algebras associated with minimal homeomorphisms of the Cantor set, Pacific Journal of Mathematics136 (1989), 329–353.
[Roy] H. Royden,Real Analysis, MacMillan, London, 1963.
[Rob] C. Robinson,Dynamical Systems, CRC Press, Boca Raton, 1995.
[Sch] S. Schwartzman,Asymptotic cycles, Annals of Mathematics66 (1957), 270–284.
[Sm] S. Smale,Differentiable dynamical systems, Bulletin of the American Mathematical Society73 (1967), 747–817.
[T] P. Trow,Degrees of constant to one factor maps, Proceedings of the American Mathematical Society103 (1988), 184–188.
[V] A. M. Verŝik,A theorem on periodical Markov approximation in ergodic theory, Journal of Soviet Mathematics28 (1985), 667–673.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boyle, M., Handelman, D. Orbit equivalence, flow equivalence and ordered cohomology. Israel J. Math. 95, 169–210 (1996). https://doi.org/10.1007/BF02761039
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02761039