ON DIGIT FREQUENCIES IN beta-EXPANSIONS

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. (AU)