On digit frequencies in {\beta}-expansions
Dynamical Systems
2014-09-02 v2 Number Theory
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.
Cite
@article{arxiv.1308.4437,
title = {On digit frequencies in {\beta}-expansions},
author = {Philip Boyland and André de Carvalho and Toby Hall},
journal= {arXiv preprint arXiv:1308.4437},
year = {2014}
}
Comments
39 pages, 10 figures. Modifications from version 1: digit frequency set now proved compact rather than having closure in its definition. Sections 4.3 and 4.4 are new