Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T16:58:23.826Z Has data issue: false hasContentIssue false
  • English
  • Français

The homoclinic and heteroclinic C*-algebras of a generalized one-dimensional solenoid

Published online by Cambridge University Press:  29 June 2009

KLAUS THOMSEN
KLAUS THOMSEN*
Affiliation:
Institut for matematiske fag, Ny Munkegade, 8000 Aarhus C, Denmark (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

D. Ruelle and I. Putnam have constructed three C*-algebras from the homoclinic and heteroclinic structure of a Smale space. This paper gives gives a complete description of these algebras when the Smale space is one of the generalized one-dimensional solenoids studied by R. Williams and I. Yi.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 263 - 308
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aarts, J. M. and Oversteegen, L.. Matchbox Manifolds, Continua (Cincinatti, OH, 1994) (Lecture Notes in Pure and Applied Mathemtics, 170). Dekker, New York, 1995, pp. 3–14.Google Scholar
[2]Bratteli, O., Evans, D. E. and Kishimoto, A.. Crossed products of totally disconnected spaces by ℤ2*ℤ2. Ergod. Th. & Dynam. Sys. 13 (1993), 445–484.Google Scholar
[3]Blackadar, B.. K-theory for Operator Algebras. Springer, New York, 1986.Google Scholar
[4]Bratteli, O., Jorgensen, P. E. T., Kim, K. H. and Roush, F.. Computation of isomorphism invariants for stationary dimension groups. Ergod. Th. & Dynam. Sys. 22 (2002), 99–127.Google Scholar
[5]Brown, L.. Stable isomorphism of hereditary subalgebras of C *-algebras. Pacific J. Math. 71 (1977), 335–348.Google Scholar
[6]Brown, L. and Pedersen, G. K.. C *-algebras of real rank zero. J. Funct. Anal. 99 (1991), 132–149.Google Scholar
[7]Cuntz, J. and Krieger, W.. A class of C *-algebras and topological Markov chains. Invent. Math. 56 (1980), 25–268.CrossRefGoogle Scholar
[8]Effros, E.. Dimensions and C *-Algebras (CBMS Regional Conference Series in Mathematics, 46). American Mathematical Society, Providence, RI, 1981.CrossRefGoogle Scholar
[9]Elliott, G.. A classification of certain simple C *-algebras, II. J. Ramanujan Math. Soc. 12 (1997), 97–134.Google Scholar
[10]Elliott, G.. On the classification of C *-algebras of real rank zero. J. Reine Angew. Math. 443 (1993), 179–219.Google Scholar
[11]Elliott, G. and Gong, G.. On the classification of C *-algebras of real rank zero, II. Ann. of Math. (2) 144 (1996), 497–610.Google Scholar
[12]Elliott, G., Gong, G. and Li, L.. On the classification of simple inductive limit C *-algebras. II. The isomorphism theorem. Invent. Math. 168(2) (2007), 249–320.Google Scholar
[13]Fokkink, R.. The structure of trajectories. PhD Thesis, Technische Universiteit te Delft, 1991.Google Scholar
[14]Fuchs, L.. Abelian Groups. Pergamon Press, Oxford, 1960.Google Scholar
[15]Giordano, T., Putnam, I. and Skau, C.. Topological orbit equivalence and C *-crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
[16]Giordano, T., Putnam, I. and Skau, C.. Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys. 24(2) (2004), 441–475.Google Scholar
[17]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.Google Scholar
[18]Hjelmborg, J. v. B. and Rørdam, M.. On stability of C *-algebras. J. Funct. Anal. 155 (1998), 152–170.Google Scholar
[19]Krieger, W.. On a dimension for a class of homeomorphism groups. Math. Ann. 252 (1979/1980), 87–95.Google Scholar
[20]Krieger, W.. On dimension functions and topological Markov chains. Invent. Math. 56 (1980), 239–250.Google Scholar
[21]Lin, H.. Tracially AF C *-algebras. Trans. Amer. Math. Soc. 353 (2001), 693–722.Google Scholar
[22]Lin, H.. The tracial topological rank of C *-algbras. Proc. London Math. Soc. 83 (2001), 199–234.Google Scholar
[23]Lin, H.. Traces and simple C *-algebras with tracial topological rank zero. J. Reine Angew. Math. 568 (2004), 99–137.Google Scholar
[24]Lin, H.. Classification of simple C *-algebras of tracial rank zero. Duke Math. J. 125 (2004), 91–119.Google Scholar
[25]Muhly, P., Renault, J. and Williams, D.. Equivalence and isomorphism for groupoid C *-algebras. J. Operator Theory 17 (1987), 3–22.Google Scholar
[26]Natsume, T.. On K *(C *(SL 2(ℤ))). J. Operator Theory 13 (1985), 103–118.Google Scholar
[27]Osaka, H. and Phillips, N. C.. Crossed products by finite groups with the Rokhlin property. Preprint, 2007, arXiv:math/0704.3651v [math.OA].Google Scholar
[28]Phillips, N. C.. Crossed products of the cantor set by free minimal actions of ℤd. Comm. Math. Phys. 256 (2005), 1–42.Google Scholar
[29]Phillips, N. C.. Every simple higher dimensional noncommutative torus is an AT algebra. Preprint, September, 2006.Google Scholar
[30]Phillips, N. C.. Real rank and property (SP) for direct limits of recursive subhomogeneous algebras, Trans. Amer. Math. Soc. to appear.Google Scholar
[31]Phillips, N. C.. The tracial Rokhlin property for actions of finite groups on C *-algebras. Preprint, 2006, arXiv:math/0609782v1 [math.OA].Google Scholar
[32]Phillips, N. C.. Finite cyclic group actions with the tracial Rokhlin property. Preprint, 2006, arXiv:math/0609785v1 [math.OA].Google Scholar
[33]Putnam, I.. C *-algebras from Smale spaces. Canad. J. Math. 48 (1996), 175–195.Google Scholar
[34]Putnam, I. and Spielberg, J.. The structure of C *-algebras associated with hyperbolic dynamical systems. J. Funct. Anal. 163 (1999), 279–299.Google Scholar
[35]Renault, J.. A Groupoid Approach to C *-algebras (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.Google Scholar
[36]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison-Wesley, Reading, MA, 1978.Google Scholar
[37]Ruelle, D.. Non-commutative algebras for hyperbolic diffeomorphisms. Invent. Math. 93 (1988), 1–13.Google Scholar
[38]Thomsen, K.. C *-algebras of homoclinic and heteroclinic structure in expansive dynamics. Preprint, IMF Aarhus University, 2007.Google Scholar
[39]Thomsen, K.. Traces, unitary characters and crossed products by ℤ. Publ. Res. Inst. Math. Sci. 31 (1995), 1011–1029.Google Scholar
[40]Thomsen, K.. On the K-theory and the E-theory of amalgamated free products of C *-algebras. J. Funct. Anal. 201 (2003), 30–56.Google Scholar
[41]Williams, R. F.. One-dimensional non-wandering sets. Topology 6 (1967), 473–487.CrossRefGoogle Scholar
[42]Williams, R. F.. Classification of one dimensional attractors. Proc. Sympos. Pure Math. 14 (1970), 341–361.Google Scholar
[43]Yi, I.. Canonical symbolic dynamics for one-dimensional generalized solenoids. Trans. Amer. Math. Soc. 353 (2001), 3741–3767.Google Scholar
[44]Yi, I.. Ordered group invariants for one-dimensional spaces. Fund. Math. 170 (2001), 267–286.Google Scholar
[45]Yi, I.. Ordered group invariants for nonorientable one-dimensional generalized solenoids. Proc. Amer. Math. Soc. 131 (2003), 1273–1282.Google Scholar
[46]Yi, I.. K-theory of C *-algebras from one-dimensional generalized solenoids. J. Operator Theory 50 (2003), 283–295.Google Scholar
[47]Yi, I.. Bratteli–Vershik systems for one-dimensional generalized solenoids. Houston J. Math. 30 (2004), 691–704.Google Scholar