Dynamical Systems
Cantor aperiodic systems and Bratteli diagrams
Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 43-46.

We prove that every Cantor aperiodic system is homeomorphic to the Vershik map acting on the space of infinite paths of an ordered Bratteli diagram and give several corollaries of this result.

Nous démontrons que chaque système de Cantor apériodique est homéomorphe à une application de Vershik agissant dans l'espace de chemins infinis d'un diagramme de Bratteli ordonné et donnons quelques applications de ce résultat.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.10.024
Medynets, Konstantin 1

1 Institute for Low Temperature Physics, 61103 Kharkov, Ukraine
@article{CRMATH_2006__342_1_43_0,
     author = {Medynets, Konstantin},
     title = {Cantor aperiodic systems and {Bratteli} diagrams},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {43--46},
     publisher = {Elsevier},
     volume = {342},
     number = {1},
     year = {2006},
     doi = {10.1016/j.crma.2005.10.024},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.10.024/}
}
TY  - JOUR
AU  - Medynets, Konstantin
TI  - Cantor aperiodic systems and Bratteli diagrams
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 43
EP  - 46
VL  - 342
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.10.024/
DO  - 10.1016/j.crma.2005.10.024
LA  - en
ID  - CRMATH_2006__342_1_43_0
ER  - 
%0 Journal Article
%A Medynets, Konstantin
%T Cantor aperiodic systems and Bratteli diagrams
%J Comptes Rendus. Mathématique
%D 2006
%P 43-46
%V 342
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.10.024/
%R 10.1016/j.crma.2005.10.024
%G en
%F CRMATH_2006__342_1_43_0
Medynets, Konstantin. Cantor aperiodic systems and Bratteli diagrams. Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 43-46. doi : 10.1016/j.crma.2005.10.024. http://www.numdam.org/articles/10.1016/j.crma.2005.10.024/

[1] Bezuglyi, S.; Dooley, A.H.; Medynets, K. The Rokhlin lemma for homeomorphisms of a Cantor set, Proc. Amer. Math. Soc., Volume 133 (2005), pp. 2957-2964

[2] Dougherty, R.; Jackson, S.; Kechris, A.S. The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc., Volume 341 (1994) no. 1, pp. 193-225

[3] Durand, F.; Host, B.; Skau, C. Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems, Volume 19 (1999), pp. 953-993

[4] Giordano, T.; Putnam, I.; Skau, C. Topological orbit equivalence and C-crossed products, J. Reine Angew. Math., Volume 469 (1995), pp. 51-111

[5] Giordano, T.; Putnam, I.; Skau, C. Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory Dynam. Systems, Volume 24 (2004), pp. 441-475

[6] Glasner, E.; Weiss, B. Weak orbit equivalence of Cantor minimal systems, Int. J. Math., Volume 6 (1995) no. 4, pp. 559-579

[7] Herman, R.H.; Putnam, I.; Skau, C. Ordered Bratteli diagram, dimension groups, and topological dynamics, Int. J. Math., Volume 3 (1992), pp. 827-864

[8] Matui, H. Topological orbit equivalence of locally compact Cantor minimal systems, Ergodic Theory Dynam. Systems, Volume 22 (2002) no. 6, pp. 1871-1903

Cited by Sources: