Beta-shifts without typical periodic optimization
Abstract
We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have typical periodic optimization (TPO): on each such beta-shift there is a non-empty open set of Lipschitz functions whose unique maximizing measure is the Sturmian measure, which is not supported on a periodic orbit. 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 proofs is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional. We also show that the set of β>1 for which the beta-shift or the beta-transformation does not have TPO is dense in (1,∞), but has zero packing dimension. The density is proved by a tuning construction, analogous to Douady--Hubbard tuning in the Mandelbrot set, which produces non-Sturmian subshifts with the same rigidity property. Finally, we show that every beta-shift has typical zero-entropy optimization: for an open dense set of Lipschitz functions, the maximizing measure is unique and its support has zero topological entropy.
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BibTeX
@misc{huang2026betashifts,
title = {{Beta-shifts without typical periodic optimization}},
author = {Wen Huang and Oliver Jenkinson and Leiye Xu and Yiwei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03619},
eprint = {2610.03619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03619},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beta-shifts without typical periodic optimization
AU - Wen Huang
AU - Oliver Jenkinson
AU - Leiye Xu
AU - Yiwei Zhang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.03619
UR - https://arxiv.org/abs/2610.03619
AB - We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have typical periodic optimization (TPO): on each such beta-shift there is a non-empty open set of Lipschitz functions whose unique maximizing measure is the Sturmian measure, which is not supported on a periodic orbit. 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 proofs is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional. We also show that the set of β>1 for which the beta-shift or the beta-transformation does not have TPO is dense in (1,∞), but has zero packing dimension. The density is proved by a tuning construction, analogous to Douady--Hubbard tuning in the Mandelbrot set, which produces non-Sturmian subshifts with the same rigidity property. Finally, we show that every beta-shift has typical zero-entropy optimization: for an open dense set of Lipschitz functions, the maximizing measure is unique and its support has zero topological entropy.
ER -