We prove the equivalence of two possible definitions of rotational interval exchange transformations: by the first definition, this is the first return map for the rotation of a circle onto a union of finitely many circle arcs, whereas by the second definition, this is an interval exchange with a scheme (in a sense of interval rearrangement ensemble) whose dual is also an interval exchange scheme.
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A. Teplinsky, “Interval rearrangement ensembles,” Ukr. Math. Zh., 75, No. 2, 247–268 (2023); English translation: Ukr. Math. J., 75, No. 2, 282–304 (2023).
M. Keane, “Interval exchange transformation,” Math. Z., 141, 25–31 (1975).
W. A. Veech, “Interval exchange transformations,” Anal. Math., 33, 222–272 (1978).
G. Rauzy, “ Échanges d’intervalles et transformations induites,” Acta Arith., 34, 315–328 (1979).
M. S. Keane and G. Rauzy, “Stricte ergodicité des échanges d’intervalles,” Math. Z., 174, 203–212 (1980).
H. Masur, “Interval exchange transformations and measured foliations,” Ann. Math. (2), 115, 169–200 (1982).
W. A. Veech, “Gauss measures for transformations on the space of interval exchange maps,” Ann. Math. (2), 115, 201–242 (1982).
A. Teplinsky, “Hyperbolic horseshoe for circle diffeomorphisms with break,” Nonlin. Oscillat., 11, No. 1, 114–134 (2008).
K. Khanin and A. Teplinsky, “Renormalization horseshoe and rigidity for circle diffeomorphisms with breaks,” Comm. Math. Phys., 320, 347–377 (2013).
K. M. Khanin and E. B. Vul, “Circle homeomorphisms with weak discontinuities,” Dynamical Systems and Statistical Mechanics, Moscow, 57–98 (1991); English translation: Adv. Soviet Math., 3, American Mathematical Society, Providence, RI (1991).
K. Cunha and D. Smania, “Renormalization for piecewise smooth homeomorphisms on the circle,” Ann. Inst. H. Poincaré C Anal. Non Linéaire, 30, 441–462 (2013).
K. Cunha and D. Smania, “Rigidity for piecewise smooth homeomorphisms on the circle,” Adv. Math., 250, 193–226 (2014).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 3, pp. 447–467, March, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i3.7779.
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Teplinsky, A. Rotational Interval Exchange Transformations. Ukr Math J 76, 501–521 (2024). https://doi.org/10.1007/s11253-024-02334-7
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DOI: https://doi.org/10.1007/s11253-024-02334-7