Abstract
We study the group of interval exchange transformations. Let T be an m-interval exchange transformation. By the rank of T we mean the dimension of the ℚ-vector space spanned by the lengths of the exchanged subintervals. We prove that if T satisfies Keane’s infinite distinct orbit condition and rank(T) > 1 + [m/2], then the only interval exchange transformations which commute with T are its powers.
In the case that T is a minimal 3-interval exchange transformation, we prove a more precise result: T has a trivial centralizer in the group of interval exchange transformations if and only if T satisfies the infinite distinct orbit condition.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Bernazzani, Most interval exchanges have no roots, Journal of Modern Dynamics 11 (2017), 249–262.
M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory and Dynamical Systems 8 (1988), 379–394.
A. del Junco, A family of counterexamples in ergodic theory, Israel Journal of Mathematics 44 (1983), 160–188.
M. Keane, Interval exchange transformations, Mathematische Zeitschrift 141 (1975), 25–31.
A. Nogueira and D. Rudolph, Topological weak-mixing of interval exchange maps, Ergodic Theory and Dynamical Systems 17 (1997), 1183–1209.
C. Novak, Discontinuity growth of interval exchange maps, Journal of Modern Dynamics 3 (2009), 379–405.
M. Viana, Ergodic theory of interval exchange maps, Revista Matemática Complutense 19 (2006), 7–100.
Acknowledgments
I would like to thank Michael Boshernitzan for encouraging me to investigate this topic, for reading many drafts of this paper, and for indicating how to prove Proposition 6.3. I would also like to thank the referee for reading the paper carefully and making several helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bernazzani, D. Centralizers in the group of interval exchange transformations. Isr. J. Math. 233, 29–48 (2019). https://doi.org/10.1007/s11856-019-1902-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1902-6