Abstract
LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1 [0,β] (x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals.
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This paper was prepared while I was very graciously hosted by the Centro de Investigacion y Estudios Avanzados, Mexico City.
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Oren, I. Ergodicity of cylinder flows arising from irregularities of distribution. Israel J. Math. 44, 127–138 (1983). https://doi.org/10.1007/BF02760616
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DOI: https://doi.org/10.1007/BF02760616