  Explicit formulas for complexity and unique invariant measure of the
period-doubling subshift can be derived from those for the Thue-Morse subshift,
obtained by Brlek, De Luca and Varricchio, and Dekking. In this note we give
direct proofs based on combinatorial properties of the period-doubling
sequence. We also derive explicit formulas for correlation integral and other
recurrence characteristics of the period-doubling subshift. As a corollary we
obtain that the determinism of this subshift converges to 1 as the distance
threshold approaches 0.
