Hausdorff dimension of unique beta expansions

Given an integer N >= 2 and a real number beta > 1, let Gamma(beta,N) be the set of all x = Sigma(infinity)(i = 1) d(i)/beta(i) with d(i) is an element of{0, 1, ... , N - 1} for all i >= 1. The infinite sequence (d(i)) is called a beta-expansion of x. Let U-beta,U-N be the set of all x's in Gamma(beta,N) which have unique beta-expansions. We give explicit formula of the Hausdorff dimension of U-beta,U-N for beta in any admissible interval [beta(L), beta(U)], where beta(L) is a purely Parry number while beta(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-beta,U-N for almost every beta > 1. In particular, this improves the main results of Gabor Kallos (1999, 2001). Moreover, we find that the dimension function f (beta) = dim(H) U-beta,U-N fluctuates frequently for beta is an element of(1, N).