Year: 2026
Type: preprint
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 $(1,\infty)$, 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220390819