  We consider the 1-equivariant energy critical wave maps problem with
two-sphere target. Using a method based on matched asymptotic expansions, we
construct infinite time relaxation, blow-up, and intermediate types of
solutions that have topological degree one. More precisely, for a symbol class
of admissible, time-dependent length scales, we construct solutions which can
be decomposed as a ground state harmonic map (soliton) re-scaled by an
admissible length scale, plus radiation, and small corrections which vanish (in
a suitable sense) as time approaches infinity. Our class of admissible length
scales includes positive and negative powers of t, with exponents sufficiently
small in absolute value. In addition, we obtain solutions with soliton length
scale undergoing damped or undamped oscillations in a bounded set, or
undergoing unbounded oscillations, for all sufficiently large t.
