On points without universal expansions

Abstract

Let 1<β<2. Given any x∈[0,(β−1)−1], a sequence (an)∈{0,1}N is called a β-expansion of x if x=∑∞n=1anβ−n. For any k≥1 and any (b1⋯bk)∈{0,1}k, if there exists some k0 such that ak0+1ak0+2⋯ak0+k=b1⋯bk, then we call (an) a universal β-expansion of x. Sidorov (2003) and Dajani and de Vries (2007) proved that for any 1<β<2, Lebesgue almost every point has uncountably many universal β-expansions. In this paper we consider the set Vβ of points without universal β-expansions. For any n≥2, let βn be the n-bonacci number, i.e., βn=βn−1+βn−2+⋯+β+1. Then dimH(Vβn)=1, where dimH denotes the Hausdorff dimension. Similar results are available for some other algebraic numbers. As a corollary, we give some results on the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper [Nonlinearity 30 (2017)].