Journal of Mathematical Physics, Volume 62, Issue 4, April 2021. Considering [math] skew-product maps over circle rotations, we prove that a renormalization transformation associated with the golden mean α* has a nontrivial periodic orbit of length 3. We also present some numerical results, including evidence that this period 3 describes scaling properties of the Hofstadter butterfly near the top of the spectrum at α* and scaling properties of the generalized eigenfunction for this energy.

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