ENTROPY, TOPOLOGICAL TRANSITIVITY, AND DIMENSIONAL PROPERTIES OF UNIQUE q-EXPANSIONS

Let M be a positive integer and q is an element of (1, M + 1]. We consider expansions of real numbers in base q over the alphabet {0,..., M}. In particular, we study the set u(q) of real numbers with a unique q-expansion, and the set U-q of corresponding sequences. It was shown by Komornik, Kong, and Li that the function H, which associates to each q is an element of (1, M+1] the topological entropy of u(q), is a Devil's staircase. In this paper we explicitly determine the plateaus of H, and characterize the bifurcation set epsilon of q's where the function H is not locally constant. Moreover, we show that epsilon is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift (V-q, sigma), which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of u(q) coincide for all q is an element of (1, M +1].