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Topological weak-mixing of interval exchange maps. (English) Zbl 0958.37010
Summary: An interval map with only one discontinuity is isomorphic to a rotation of the circle, and has continuous eigenfunctions. We show here that for almost every choice of lengths of the intervals, this is the only way an irreducible interval exchange can have a somewhere continuous eigenfunction. We show slightly more, considering certain towers over the interval exchange, showing that outside of a set of choices for interval lengths of measure zero these have a somewhere continuous eigenfunction only if they are isomorphic to either a rotation, or a tower of constant height over an interval exchange.

37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37A30 Ergodic theorems, spectral theory, Markov operators
54H20 Topological dynamics (MSC2010)
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