Abstract
We construct interval exchange transformations on four intervals satisfying a strong irrationality condition and having exactly two ergodic invariant probability measures. This shows that although Kronecker’s theorem remains true for interval exchange transformations, the Weyl equidistribution theorem is false even under the strongest irrationality assumptions.
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References
M. Keane,Interval exchange transformations, Math. Z.141 (1975), 25–31.
H. B. Keynes and D. Newton,A “minimal”, non-uniquely ergodic interval exchange transformation, to appear.
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Keane, M. Non-ergodic interval exchange transformations. Israel J. Math. 26, 188–196 (1977). https://doi.org/10.1007/BF03007668
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DOI: https://doi.org/10.1007/BF03007668