Abstract
An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds.
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A. V. Kočergin,On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus, Soviet Math. Dokl.13 (1972), 949–952. Translated from Russian.
A. N. Zemlyakov and A. B. Katok,Topological transitivity of billiards in polygons, Math. Notes18 (1975), 760–764; errata20 (1976), 1051. Translated from Russian.
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The author is partially supported by grant NSF MCS 78-15278.
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Katok, A. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35, 301–310 (1980). https://doi.org/10.1007/BF02760655
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DOI: https://doi.org/10.1007/BF02760655