Univoque bases of real numbers: Local dimension, Devil's staircase and isolated points

Given a positive integer M and a real number x>0, let U(x) be the set of all bases q∈(1,M+1] for which there exists a unique sequence (d_i)=d₁d₂… with each digit d_i∈{0,1,…,M} satisfying [Formula presented] The sequence (d_i) is called a q-expansion of x. In this paper we investigate the local dimension of U(x) and prove a ‘variation principle’ for unique non-integer base expansions. We also determine the critical values of U(x) such that when x passes the first critical value the set U(x) changes from a set with positive Hausdorff dimension to a countable set, and when x passes the second critical value the set U(x) changes from an infinite set to a singleton. Denote by U(x) the set of all unique q-expansions of x for q∈U(x). We give the Hausdorff dimension of U(x) and show that the dimensional function x↦dim_H⁡U(x) is a non-increasing Devil's staircase. Finally, we investigate the topological structure of U(x). Although the set U(1) has no isolated points, we prove that for typical x>0 the set U(x) contains isolated points.