Structure of three-interval exchange transformations II: a combinatorial description of the tranjectories

Journal d'Analyse Mathematique

Volumn 89, Issue 1, 2003, Pages 239-276

  Ferenczi, Sébastien ^a,b   Holton, Charles ^c   Zamboni, Luca Q ^d  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b Aix Marseille University   (France)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d UNIVERSITY OF NORTH TEXAS   (United States)

Author keywords

Exchange Transformation; Interval Exchange Transformation; Left Endpoint; Sturmian Sequence; Symbolic Sequence

Indexed keywords

EID: 7544231828     PISSN: 00217670     EISSN: 15658538     Source Type: Journal    
DOI: 10.1007/BF02893083     Document Type: Article

References (23)

1. V. I. Arnold,Small denominators and problems of stability of motion in classical and celestial mechanics, Uspekhi Mat. Nauk18 (1963), 91–192; translated in: Russian Math. Surveys18 (1963), 86–194.
2. P. Arnoux and G. Rauzy,Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France119 (1991), 199–215.
3. V. Berthé, N. Chekhova and S. Ferenczi,Covering numbers: arithmetics and dynamics for rotations and interval exchanges, J. Analyse Math.79 (1999), 1–31.
4. M. Boshernitzan and C. Carroll,An extension of Lagrange's theorem to interval exchange transformations over quadratic fields, J. Analyse Math.72 (1997), 21–44. DOI: 10.1007/BF02843152
5. J. Cassaigne,Complexité et facteurs spéciaux, Bull. Belg. Math. Soc.4 (1997), 67–88.
6. E. M. Coven and G. A. Hedlund,Sequences with minimal block growth, Math. Systems Theory7 (1972), 138–153. DOI: 10.1007/BF01762232
7. A. del Junco,A family of counterexamples in ergodic theory, Israel J. Math.44 (1983), 160–188. DOI: 10.1007/BF02760618
8. G. Didier,Échanges de trois intervalles et suites sturmiennes, J. Théor. Nombres Bordeaux9 (1997), 463–478.
9. X. Droubay, J. Justin and G. Pirillo,Epi-Sturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci.255 (2001), 539–553. DOI: 10.1016/S0304-3975(99)00320-5
10. F. Durand,A characterization of substitutive sequences using return words, Discrete Math.179 (1998), 89–101. DOI: 10.1016/S0012-365X(97)00029-0
11. S. Ferenczi,Rank and symbolic complexity, Ergodic Theory Dynam. Systems16 (1996), 663–682.
12. S. Ferenczi,Systems of finite rank, Colloq. Math.73 (1997), 35–65.
13. S. Ferenczi, C. Holton and L. Zamboni,The structure of three-interval exchange transformations I: an arithmetic study, Ann. Inst. Fourier51 (2001), 861–901.
14. S. Ferenczi, C. Holton and L. Zamboni,The structure of three-interval exchange transformations III: ergodic and spectral properties; preprint 2001–26, http://iml.univ-mrs.fr/editions/preprint2001/preprint2001.html.
15. C. Holton and L. Zamboni,Descendants of primitive substitutions, Theoret. Comput. Systems32 (1999), 133–157. DOI: 10.1007/s002240000114
16. A. B. Katok and A. M. Stepin,Approximations in ergodic theory, Uspekhi Mat. Nauk22 (1967), 81–106; translated in: Russian Math. Surveys22 (1967), 76–102.
17. M. S. Keane,Interval exchange transformations, Math. Z.141 (1975), 25–31. DOI: 10.1007/BF01236981
18. M. Morse and G. A. Hedlund,Symbolic dynamics II. Sturmian trajectories, Amer. J. Math.62 (1940), 1–42. DOI: 10.2307/2371431
19. G. Rauzy,Échanges d'intervalles et transformations induites, Acta Arith.34 (1979), 315–328.
20. R. Risley and L. Q. Zamboni,A generalization of Sturmian sequences: combinatorial structure and transcendence, Acta Arith.95 (2000), 167–184.
21. M.-L. Santini-Bouchard,Échanges de trois intervalles, et suites minimales, Theoret. Comput. Sci.174 (1997), 171–191. DOI: 10.1016/S0304-3975(96)00023-0
22. W. A. Veech,The metric theory of interval exchange transformations I, II, III, Amer. J. Math.106 (1984), 1331–1421. DOI: 10.2307/2374396
23. A. M. Vershik and A. N. Livshits,Adic models of ergodic transformations, spectral theory, substitutions, and related topics, Adv. Soviet Math.9 (1992), 185–204.