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Infinite interval exchange transformations from shifts

Published online by Cambridge University Press:  12 May 2016

LUIS-MIGUEL LOPEZ  and
PHILIPPE NARBEL
LUIS-MIGUEL LOPEZ
Affiliation:
Tokyo University of Social Welfare, Isesaki, 372-0831 Gunma, Japan email [email protected]
PHILIPPE NARBEL
Affiliation:
LaBRI, University of Bordeaux, 33405 Talence, France email [email protected]
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Abstract

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We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 6 , September 2017 , pp. 1935 - 1965
Copyright
© Cambridge University Press, 2016 

References

Avgustinovich, S. V., Frid, A. E. and Puzynina, S.. Canonical representatives of morphic permutations. WORDS’15 (Lecture Notes in Computer Science, 9304) . Springer, Berlin, 2015, pp. 59–72.Google Scholar
Arnoux, P., Ornstein, D. and Weiss, B.. Cutting and stacking, interval exchanges and geometric models. Israel J. Math. 50(1–2) (1985), 160–168.CrossRefGoogle Scholar
Cassaigne, J.. Special factors of sequences with linear subword complexity. Developments in Language Theory, II (Magdeburg, 1995). World Sci. Publ., River Edge, NJ, 1996, pp. 25–34.Google Scholar
Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Y. G.. Ergodic Theory Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences, 245) . Springer, New York, 1982.Google Scholar
Cassaigne, J. and Nicolas, F.. Factor complexity. Combinatorics, Automata and Number Theory (Encyclopedia Mathematics and its Applications, 135) . Eds. Berthé, V. and Rigo, M.. Cambridge Unversity Press, Cambridge, 2010, pp. 163–247.CrossRefGoogle Scholar
Dekking, F. M.. On the Thue–Morse measure. Acta Univ. Carolin. Math. Phys. 33(2) (1992), 35–40.Google Scholar
Ehrenfeucht, A. and Rozenberg, G.. Repetitions of subwords in D0L languages. Inform. Control 59 (1983), 13–35.CrossRefGoogle Scholar
Pytheas Fogg, N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794) . Eds. Berthé, V., Ferenczi, S., Mauduit, C. and Siegel, A.. Springer, Berlin, 2002.CrossRefGoogle Scholar
Hasselblatt, B. and Katok, A.. Principal structures. Handbook of Dynamical Systems. Vol. 1A. North-Holland, Amsterdam, 2002, pp. 1–203.Google Scholar
Hooper, W. P.. The invariant measures of some infinite interval exchange maps. Geom. Topol. 19 (2015), 1895–2038.CrossRefGoogle Scholar
Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
Kitchens, B. P.. Symbolic Dynamics. Springer, Berlin, 1998.CrossRefGoogle Scholar
Kolyada, S., Snoha, L. and Trofimchuk, S.. Noninvertible minimal maps. Fund. Math. 168(2) (2001), 141–163.CrossRefGoogle Scholar
Lind, D. and Marcus, B.. Symbolic Dynamics and Coding. Cambridge Unversity Press, Cambridge, 1995.CrossRefGoogle Scholar
Makarov, M. A.. On an infinite permutation similar to the Thue–Morse word. Discrete Math. 309(23–24) (2009), 6641–6643.CrossRefGoogle Scholar
Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics I. Amer. J. Math. 60 (1938), 815–866.CrossRefGoogle Scholar
Pansiot, J.-J.. Complexité des facteurs des mots infinis engendrés par morphismes itérés. ICALP’84 (Lecture Notes in Computer Science, 172) . Springer, Berlin, 1984, pp. 380–389.Google Scholar
Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.CrossRefGoogle Scholar
Quéffelec, M.. Substitution Dynamical Systems—Spectral Analysis (Lecture Notes in Mathematics, 1294) . 2nd edn. Springer, Berlin, 2010.CrossRefGoogle Scholar
Thue, A.. Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen [On the relative position of equal parts in certain sequences of symbols]. Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl. 10 (1912), 1–67. Reprinted in Selected Mathematical Papers of Axel Thue. Ed. T. Nagell et al. Universitetsforlaget, Oslo, 1977, pp. 413–478.Google Scholar