Abstract

We introduce the Thue-Morse sequence with many examples, and then give the definition and properties of automatic sequences in general. We talk about the link between automaticity and algebraicity, the results for power series in GF(q)[[x]] due to Christol, Kamae, Mendès-France and Rauzy, for expansion of real numbers due to Adamczewski and Bugeaud, and for real continued fractions due to Bugeaud. There seems to be a missing piece of the puzzle – What about power series that are continued fractions defined by automatic sequences? We attempt to answer this question by looking at the typical example, the Thue-Morse sequence.