# Problem 81

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Geometric proof of unique ergodicity for irrational rotation of $\mathbb S^1$

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### I guess that the question is

I guess that the question is motivated by the parallel between the horocyclic flow and irrational rotations of the circle. The original proof of the  unique ergodicity of the horocycle flow by Furstenberg used harmonic analysis, then R. Bowen and B. Marcus gave a proof of a more geometrical flavor, using the equicontinuity of the push-backward of the stable leaves.

There is of course a proof of the unique ergodicity of rotations, and more generally of 1-Lipschitz transformations, in the spirit of R. Bowen and B. Marcus. Consider a 1-Lipschitz transformation of a compact metric space. The Birkhoff sums associated to a continuous observable forms an equicontinuous family of transformations,  hence by Ascoli theorem, is compact with respect to the uniform norm.  Hence any accumulation point is both continuous and invariant (by the ergodic theorem). If the transformation is transitive, we deduce that the  limit is constant and unique ergodicity follows. The argument somehow works even if the underlying space is not compact.

There is also a way to prove the ergodicity of an irrational rotation by mimicking the Hopf argument, this time applied to the dual mean stable leaves $W_{dual}^s(x) = \{y \mid \forall f Lipschitz, \ {1\over n} (S_n(f)(x) - S_n(f)(y)) \rightarrow 0 \}$, which happen to be a "geometric" realization of the ergodic components of the transformation.