Hausdorff dimension of unique beta expansions

Given an integer N ≥ 2 and a real number β > 1, let Γ_β,N be the set of all x = Σ_i=1^∞ d_i/βⁱ with d_i ∈ {0, 1, ⋯, N - 1} for all i ≥ 1. The infinite sequence (d_i) is called a β-expansion of x . Let U_β,N be the set of all x's in Γ_β,N which have unique β-expansions. We give explicit formula of the Hausdorff dimension of U_β,N for β in any admissible interval [β_L, β_U], where β_L is a purely Parry number while β_U is a transcendental number whose quasi-greedy expansion of 1 is related to the classical Thue-Morse sequence. This allows us to calculate the Hausdorff dimension of U_β,N for almost every β > 1. In particular, this improves the main results of Gábor Kallós (1999, 2001). Moreover, we find that the dimension function f(β) = dim_H U_βN fluctuates frequently for β ∈ (1, N).