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 $\beta >1$ 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.

arXiv comment54 pages. Results added showing that the set of non-TPO parameters is dense and of zero packing dimension, and that every beta-shift has typical zero-entropy optimization. Title change to reflect the new content