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On Periods of Interval Exchange Transformations

Author

Listed:
  • Jose S. Cánovas

    (Department of Applied Mathematics and Statistics, Technical University of Cartagena, 30202 Cartagena, Spain)

  • Antonio Linero Bas

    (Department of Mathematics, University of Murcia, 30100 Murcia, Spain)

  • Gabriel Soler López

    (Department of Applied Mathematics and Statistics, Technical University of Cartagena, 30202 Cartagena, Spain)

Abstract

In this paper, we study the periods of interval exchange transformations. First, we characterize the periods of interval exchange transformations with one discontinuity. In particular, we prove that there is no forcing between periods of maps with two branches of continuity. This characterization is a partial solution to a problem by Misiurewicz. In particular, a periodic structure is not possible for a family of the maps with one point of discontinuity in which the monotonicity changes. Second, we study a similar problem for interval exchange transformations with two discontinuities. Here we classify these maps in several classes such that two maps in the same class have the same periods. Finally, we study the set of periods for two categories, obtaining partial results that prove that the characterization of the periods in each class is not an easy problem.

Suggested Citation

  • Jose S. Cánovas & Antonio Linero Bas & Gabriel Soler López, 2022. "On Periods of Interval Exchange Transformations," Mathematics, MDPI, vol. 10(9), pages 1-32, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1487-:d:805555
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    References listed on IDEAS

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    1. A. Douglas Stone, 2010. "Chaotic billiard lasers," Nature, Nature, vol. 465(7299), pages 696-697, June.
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