  We revisit a technique of S. Lehr on automata and use it to prove old and new
results in a simple way. We give a very simple proof of the 1986 theorem of
Honkala that it is decidable whether a given k-automatic sequence is ultimately
periodic. We prove that it is decidable whether a given k-automatic sequence is
overlap-free (or squareefree, or cubefree, etc.) We prove that the
lexicographically least sequence in the orbit closure of a k-automatic sequence
is k-automatic, and use this last result to show that several related
quantities, such as the critical exponent, irrationality measure, and
recurrence quotient for Sturmian words with slope alpha, have automatic
continued fraction expansions if alpha does.
