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arXiv · 2610.03619

Sturmian beta-shifts do not have typical periodic optimization

Wen Huang·Oliver Jenkinson·Leiye Xu·Yiwei Zhang
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Abstract

A shift space is said to have typical periodic optimization (TPO) if the set of Lipschitz functions whose unique maximizing measure is supported on a periodic orbit contains an open dense subset of the space of Lipschitz functions. We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have TPO: on each such beta-shift there is a non-empty open set of Lipschitz functions, all of which have the Sturmian measure as their unique maximizing measure. These are the first known examples of beta-shifts without TPO, and, since they have the specification property, the first known examples of shift spaces with specification but without TPO. The corresponding beta-transformations do not have TPO in any space of Hölder functions on the interval. The main ingredient in the proof of these results is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional.

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Wen Huang, Oliver Jenkinson, Leiye Xu, Yiwei Zhang. 2026-10-02. Sturmian beta-shifts do not have typical periodic optimization. https://arxiv.org/abs/2610.03619

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