Critical exponents of balanced words

Over a binary alphabet it is well-known that the aperiodic balanced words are exactly the Sturmian words. The repetitions in Sturmian words are well-understood. In particular, there is a formula for the critical exponent (supremum of exponents e such that $x^e$ is a factor for some word x) of a Sturmian word. It is known that the Fibonacci word has the least critical exponent over all Sturmian words and this value is $(5+\sqrt{5})/2$. However, little is known about the critical exponents of balanced words over larger alphabets. We show that the least critical exponent among ternary balanced words is $2+\sqrt{2}/2$ and we construct a balanced word over a four-letter alphabet with critical exponent $(5+\sqrt{5})/4$. This is joint work with N. Rampersad and J. Shallit.