Bifurcation sets arising from non-integer base expansions

Abstract

Given a positive integer M and q\in(1,M+1] , let \mathcal U_q be the set of x\in[0, M/(q-1)] having a unique q -expansion: there exists a unique sequence (x_i)=x_1x_2\ldots with each x_i\in\{0,1,\ldots, M\} such that x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. Denote by \mathbf U_q the set of corresponding sequences of all points in \mathcal U_q . It is well-known that the function H: q\mapsto h(\mathbf U_q) is a Devil's staircase, where h(\mathbf U_q) denotes the topological entropy of \mathbf U_q . In this paper we give several characterizations of the bifurcation set \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. Note that \mathcal B is contained in the set { \mathcal{U}^R } of bases q\in(1,M+1] such that 1\in\mathcal U_q . By using a transversality technique we also calculate the Hausdorff dimension of the difference \mathcal U\backslash\mathcal{B} . Interestingly this quantity is always strictly between 0 and 1. When M=1 the Hausdorff dimension of \mathcal U\backslash \mathcal{B} is \frac{\log 2}{3\log \lambda^*}\approx 0.368699 , where \lambda^* is the unique root in (1, 2) of the equation x^5-x^4-x^3-2x^2+x+1=0 .