2017/06/16 by Pieter C. Allaart, Allaart, Pieter, Simon Baker +3 · 1 citation
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1706.05190
Given a positive integer M and q∈(1,M+1], let \mathcal 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=(x1)/(q)+(x2)/(q2)+(x3)/(q3)+⋯. Denote by \mathbf Uq the set of corresponding sequences of all points in \mathcal Uq. It is well-known that the function H: q↦ h(\mathbf Uq) is a Devil's staircase, where h(\mathbf Uq) denotes the topological entropy of \mathbf Uq. In this paper we give several characterizations of the bifurcation set \mathcal B:=\q∈(1,M+1]: H(p)≠ H(q)\textrm for any p≠ q\. Note that \mathcal B is contained in the set UR of bases q∈(1,M+1] such that 1∈\mathcal Uq. By using a transversality technique we also calculate the Hausdorff dimension of the difference \mathcal B\backslashUR. Interestingly this quantity is always strictly between 0 and 1. When M=1 the Hausdorff dimension of \mathcal B\backslashUR is (log 2)/(3log λ^*)≈ 0.368699, where λ^* is the unique root in (1, 2) of the equation x5-x4-x3-2x2+x+1=0.