arXiv:2610.03619v1
Sturmian beta-shifts do not have typical periodic optimization
math.DSmath.NT37A4437B1037E0511B85
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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01Statements3 reported findingsNo statement issues found
The failure of typical periodic optimization for Sturmian beta-shifts, the quantitative obstruction behind it, and the stated symbolic-to-interval transfer and classification results follow under the hypotheses in the manuscript.
Theorems A and CNo issue found
Sturmian balance creates a robust nonperiodic optimizer
PDF pp. 2–5, 10–23 · arXiv:2610.03619v1
The graph presentation identifies the return loops of the Sturmian beta-shift and bounds all zero runs, giving specification and density of periodic measures. The centered digit function is a bounded coboundary, while its Sturmian measure remains uniformly separated from every periodic measure in the Lipschitz dual distance. The explicitly constructed Lipschitz potential has that measure as its unique maximizer, and the separation estimate leaves an open neighborhood without periodic maximizing measures.
Theorems D and ENo issue found
The coefficient expansion and metric transfer retain the required regularity
PDF pp. 17–18, 23–29 · arXiv:2610.03619v1
Every Lipschitz potential is decomposed into a convergent family of cylinder indicators with exponentially decaying coefficients, and the centered transfer tails preserve the Lipschitz bound. The coding map is Lipschitz between the indicated symbolic and interval metrics; the exceptional boundary orbit contributes a controlled discrepancy whose time average vanishes. These facts give the claimed correspondence of maximizing measures and the equivalence of typical periodic optimization across the coding.
Theorem F and Corollary GNo issue found
The extension criterion and beta-transformation consequences follow
PDF pp. 29–31 · arXiv:2610.03619v1
Restriction maps preserve locking, while a distance penalty extends every locked potential and forces all maximizers back into the invariant subsystem; this proves both directions of the extension criterion. The cited beta-shift structure results then separate emergent from non-emergent parameters and yield the two cases in the final classification.
02Proofs2 reported findingsNo proof issues found
The return-loop combinatorics, discrepancy estimates, cylinder expansion, Wasserstein separation, coding comparison, and extension arguments were checked through their central estimates. The proof-critical beta-shift measure and optimization results cited from earlier work have the forms used here.
Sections 4–7No issue found in the reviewed proof
The symbolic obstruction closes quantitatively
PDF pp. 10–23 · arXiv:2610.03619v1
Balance of Sturmian words gives the exact digit-average bound. The circle-rotation coding converts this into bounded Birkhoff sums, and the cylinder expansion produces an explicit coboundary potential with controlled Lipschitz norm. For an arbitrary periodic measure, the first return at which its coding differs from the Sturmian orbit supplies a cylinder test with a uniform dual-distance lower bound. Combining this with the nonnegative gap identity gives uniqueness and the stated open obstruction.
Section 8No issue found in the reviewed proof
Coding, restriction, and penalized extension preserve optimization
PDF pp. 24–31 · arXiv:2610.03619v1
Lemma 8.1 controls coding distance both before and after an orbit crosses the endpoint ambiguity, and averaging removes the single exceptional boundary contribution. The pushforward correspondence preserves invariant and periodic measures and their integrals. Conversely, adding a sufficiently large multiple of distance to the invariant subset dominates the Lipschitz variation off that subset, so no new maximizing measure is introduced.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.