Exercises

Uniqueness of the period 2. If w = vukr with \v\ < \u\ and u is not periodic, then any appearance

of u as a subword of w must be in one of the k obvious positions. In other words, if w = v'ur1', then \v'\ = \v\ + i\u\ for some integer i. (Hint: Otherwise, u = u'u" where one of the obvious occurrences of u ends in u' or begins with u". But then apply Lemma 2.61 to conclude that u is periodic, a contradiction.)

3. If w is quasiperiodic with period u and has a subword v2 with \v\ > \u\, then v is cyclically conjugate to a power of u. (Hint: Cyclically conjugating if necessary, write v = u^v' with j maximal possible. Then v2 = u^v'u^v'. Since w is quasiperiodic, some

Then w' = uku" where u" is an initial subword of u, and matching parts shows u"u = uu".)

4. If quasiperiodic words w\, w<i with respective periods u\, 112 of lengths d\, d2 have a common subword v of length d\ +d2, then u\ is cyclically conjugate to t¿2-(Hint: Assume d\ > d2-Replacing u\ by a cyclic conjugate, write v = uiu[, where \u[\ = d2-Since v starts with a cyclic conjugate u2 of u2, one can write u\ — u\v![, for \u![\ < d2, so v starts with ü2uf{ü2] show u'{ = 0.)

5. (Ambiguity in the periodicity.) The periodic word 12111211 can also be viewed as the initial subword of a periodic word of periodicity 7.