  In this paper we initiate the study of equivariant wave maps from 2d
hyperbolic space into rotationally symmetric surfaces. This problem exhibits
markedly different phenomena than its Euclidean counterpart due to the
exponential volume growth of concentric geodesic spheres on the domain.
  In particular, when the target is the 2-sphere, we find a family of
equivariant harmonic maps indexed by a parameter that measures how far the
image of each harmonic map wraps around the sphere. These maps have energies
taking all values between zero and the energy of the unique co-rotational
Euclidean harmonic map, Q, from the Euclidean plane to the 2-sphere, given by
stereographic projection. We prove that the harmonic maps are asymptotically
stable for values of the parameter smaller than a threshold that is large
enough to allow for maps that wrap more than halfway around the sphere. Indeed,
we prove Strichartz estimates for the operator obtained by linearizing around
such a harmonic map. However, for harmonic maps with energies approaching the
Euclidean energy of Q, asymptotic stability via a perturbative argument based
on Strichartz estimates is precluded by the existence of gap eigenvalues in the
spectrum of the linearized operator.
  When the target is 2d hyperbolic space, we find a continuous family of
asymptotically stable equivariant harmonic maps with arbitrarily small and
arbitrarily large energies. This stands in sharp contrast to the corresponding
problem on Euclidean space, where all finite energy solutions scatter to zero
as time tends to infinity.
