Which beta-shifts have a largest invariant measure?

Abstract

For a given beta-shift, the lexicographic order induces a partial order (known as first-order stochastic dominance) on the collection of its shift-invariant probability measures. We characterize those beta-shifts for which this partial order has a largest element. These beta-shifts are all of finite type, and their lexicographically largest point is a periodic sequence of a particular kind: it is Sturmian (that is, its shift-orbit is combinatorially equivalent to a rotation) with weight-per-symbol either an integer, or equal to p/(ap + 1) for some a, p ⩾ 1, or equal to A + p/(p + 1) for some p ⩾ 1 and A ⩾ 2. In these cases, the largest invariant measure is precisely the unique one supported by the shift-orbit of the lexicographically largest point in the beta-shift.