Eigenvalues and simplicity of interval exchange transformations

Sébastien Ferenczi; Luca Q. Zamboni

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 3, page 361-392
  • ISSN: 0012-9593

Abstract

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For a class of d -interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech’s simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.

How to cite

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Ferenczi, Sébastien, and Zamboni, Luca Q.. "Eigenvalues and simplicity of interval exchange transformations." Annales scientifiques de l'École Normale Supérieure 44.3 (2011): 361-392. <http://eudml.org/doc/272107>.

@article{Ferenczi2011,
abstract = {For a class of $d$-interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech’s simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.},
author = {Ferenczi, Sébastien, Zamboni, Luca Q.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {interval exchanges; self-dual induction; eigenvalues; Veech simplicity},
language = {eng},
number = {3},
pages = {361-392},
publisher = {Société mathématique de France},
title = {Eigenvalues and simplicity of interval exchange transformations},
url = {http://eudml.org/doc/272107},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Ferenczi, Sébastien
AU - Zamboni, Luca Q.
TI - Eigenvalues and simplicity of interval exchange transformations
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 3
SP - 361
EP - 392
AB - For a class of $d$-interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first nontrivial examples with eigenvalues (rational or irrational), the first ever example of an exchange on more than three intervals satisfying Veech’s simplicity (though this weakening of the notion of minimal self-joinings was designed in 1982 to be satisfied by interval exchange transformations), and an unexpected example which is non uniquely ergodic, weakly mixing for one invariant ergodic measure but has rational eigenvalues for the other invariant ergodic measure.
LA - eng
KW - interval exchanges; self-dual induction; eigenvalues; Veech simplicity
UR - http://eudml.org/doc/272107
ER -

References

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