×

zbMATH — the first resource for mathematics

The Chacon transformations: Combinatorics, geometric structure, link with the systems of complexity \(2n+1\). (Les transformations de Chacon: Combinatoire, structure géométrique, lien avec les systèmes de complexité \(2n+1\).) (French) Zbl 0855.28008
The author shows that Chacon’s map is a system of complexity \((2n- 1)\), and describes the associated graph of words. The latter is further used to give a primitive form of the substitution, and a geometric representation of the transformation, as an exduction of a triadic rotation. Finally, the complexity of the simplest known weakly mixing systems is computed.

MSC:
28D05 Measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI Link Numdam EuDML
References:
[1] ARNOUX (P.) et RAUZY (G.) . - Représentation géométrique de suites de complexité 2n + 1 , Bull. Soc. Math. France., t. 119, 1991 , p. 199-215. Numdam | MR 92k:58072 | Zbl 0789.28011 · Zbl 0789.28011 · numdam:BSMF_1991__119_2_199_0 · eudml:87622
[2] CHACON (R.V.) . - Weakly mixing transformations which are not strongly mixing , Proc. Amer. Math. Soc., t. 22, 1969 , p. 559-562. MR 40 #297 | Zbl 0186.37203 · Zbl 0186.37203 · doi:10.2307/2037431
[3] DEKKING (F.M.) . - The spectrum of dynamical systems arising from substitutions of constant length , Zeit. Wahr., t. 41, 1978 , p. 221-239. MR 57 #1455 | Zbl 0348.54034 · Zbl 0348.54034 · doi:10.1007/BF00534241
[4] DEL JUNCO (A.) , RAHE (A. M.) and SWANSON (M.) . - Chacon’s automorphism has minimal self-joinings , J. Analyse Math., t. 37, 1980 , p. 276-284. MR 81j:28027 | Zbl 0445.28014 · Zbl 0445.28014 · doi:10.1007/BF02797688
[5] DEL JUNCO (A.) and RUDOLPH (D.J.) . - A rank-one, rigid, simple, prime map , Ergodic Th. Dyn. Syst. 7, t. 2, 1987 , p. 229-247. MR 88h:28016 | Zbl 0634.54020 · Zbl 0634.54020
[6] FERENCZI (S.) . - Systèmes de rang fini , Thèse d’Etat, Université d’Aix-Marseille 2, 1990 .
[7] FIELDSTEEL (A.) . - An uncountable family of prime transformations not isomorphic to their inverses , preprint, vers 1980 .
[8] HERMAN (R.H.) , PUTNAM (I.F.) and SKAU (C.F.) . - Ordered Bratteli diagrams , dimension groups and topological dynamics, International J. of Maths 3, t. 6, 1992 , p. 827-864. MR 94f:46096 | Zbl 0786.46053 · Zbl 0786.46053 · doi:10.1142/S0129167X92000382
[9] HOST (B.) . - Dimension groups and substitution dynamical systems , Prétirage du Laboratoire de Mathématiques Discrètes, 1994 .
[10] MOSSE (B.) . - Notions de reconnaissabilité pour les substitutions et complexité des suites automatiques , soumis.
[11] ORNSTEIN (D.S.) . - On the root problem in ergodic theory , Proc. of the Sixth Berkeley Symposium in Mathematical Statistics and Probability, Univ. of California Press, 1970 , p. 347-356. MR 53 #3259 | Zbl 0262.28009 · Zbl 0262.28009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.