Bifurcation sets arising from non-integer base expansions

Abstract

Given a positive integer M and q∈(1,M+1], let Uq be the set of x∈[0,M/(q−1)] having a unique q-expansion: there exists a unique sequence (xi)=x1x2… with each xi∈{0,1,…,M} such that x=x1q+x2q2+x3q3+⋯. Denote by Uq the set of corresponding sequences of all points in Uq. It is well-known that the function H:q↦h(Uq) is a Devil's staircase, where h(Uq) denotes the topological entropy of Uq. In this paper we give several characterizations of the bifurcation set B:={q∈(1,M+1]:H(p)≠H(q) for any p≠q}. Note that B is contained in the set {UR} of bases q∈(1,M+1] such that 1∈Uq. By using a transversality technique we also calculate the Hausdorff dimension of the difference U∖B. Interestingly this quantity is always strictly between 0 and 1. When M=1 the Hausdorff dimension of U∖B is log23logλ∗≈0.368699, where λ∗ is the unique root in (1,2) of the equation x5−x4−x3−2x2+x+1=0.