On digit frequencies in β-expansions

Abstract: We study the sets DF({\beta}) of digit frequencies of {\beta}-expansions of numbers in [0,1]. We show that DF({\beta}) is a compact convex set with countably many extreme points which varies continuously with {\beta}; that there is a full measure collection of non-trivial closed intervals on each of which DF({\beta}) mode locks to a constant polytope with rational vertices; and that the generic digit frequency set has infinitely many extreme points, accumulating on a single non-rational extreme point whose components are rationally independent.