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

AstractWe describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence, we obtain a complete characterization of those sequences of block complexity 2n+1 which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy.

[1]  V. Arnold.  SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .

[2]  Gérard Rauzy,et al.  Représentation géométrique de suites de complexité $2n+1$ , 1991 .

[3]  Sébastien Ferenczi,et al.  Structure of three interval exchange transformations I: an arithmetic study , 2001 .

[4]  Giuseppe Pirillo,et al.  Episturmian words and some constructions of de Luca and Rauzy , 2001, Theor. Comput. Sci..

[5]  Anatoly M. Vershik,et al.  Adic models of ergodic transformations, spectral theory, substitutions, and related topics , 1992 .

[6]  Sébastien Ferenczi,et al.  Covering numbers: Arithmetics and dynamics for rotations and interval exchanges , 1999 .

[7]  Luca Q. Zamboni,et al.  A generalization of Sturmian sequences: Combinatorial structure and transcendence , 2000 .

[8]  Ethan M. Coven,et al.  Sequences with minimal block growth II , 1973, Mathematical systems theory.

[9]  A. Katok,et al.  APPROXIMATIONS IN ERGODIC THEORY , 1967 .

[10]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[11]  Marie-Line Santini-Bouchard.  Echanges de trois intervalles et suites minimales , 1997, Theor. Comput. Sci..

[12]  W. Veech.  THE METRIC THEORY OF INTERVAL EXCHANGE TRANSFORMATIONS I. GENERIC SPECTRAL PROPERTIES , 1984 .

[13]  H. Masur.  Interval Exchange Transformations and Measured Foliations , 1982 .

[14]  A. Junco.  A family of counterexamples in ergodic theory , 1983 .

[15]  Luca Q. Zamboni,et al.  Descendants of Primitive Substitutions , 1999, Theory of Computing Systems.

[16]  S. Ferenczi.  Systems of finite rank , 1997 .

[17]  Gérard Rauzy,et al.  Échanges d'intervalles et transformations induites , 1979 .

[18]  Sébastien Ferenczi,et al.  Structure of three-interval exchange transformations III: Ergodic and spectral properties , 2004 .

[19]  M. Keane.  Interval exchange transformations , 1975 .

[20]  Sébastien Ferenczi,et al.  Rank and symbolic complexity , 1996, Ergodic Theory and Dynamical Systems.

[21]  Fabien Durand,et al.  A characterization of substitutive sequences using return words , 1998, Discret. Math..

[22]  C. R. Carroll,et al.  An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields , 1997 .

[23]  Julien Cassaigne,et al.  Complexité et facteurs spéciaux , 1997 .