Selfsimilarity in the Birkhoff Sum of the Cotangent Function

  title={Selfsimilarity in the Birkhoff Sum of the Cotangent Function},
  author={Oliver Knill},
We prove that the Birkhoff sum Sn(α) = 1 n ∑n k=1 g(kα) of g = cot(πx) and with golden ratio α converges in the sense that the sequence of functions sn(x) = S[xq2n]/q2n with Fibonacci qn converges to a self similar limiting function s(x) on [0, 1] which can be computed analytically. While for any continuous function g, the Birkhoff limiting function is s(x) = Mx by Birkhoff’s ergodic theorem, we get so examples of random variables Xn, where the limiting function of S[xn]/n → s(x) exists along… CONTINUE READING
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