f(T) gravity mimicking dynamical dark energy. Background and
  perturbation analysis
  We investigate f(T) cosmology in both the background, as well as in the
perturbation level, and we present the general formalism for reconstructing the
equivalent one-parameter family of f(T) models for any given dynamical dark
energy scenario. Despite the completely indistinguishable background behavior,
the perturbations break this degeneracy and the growth histories of all these
models differ from one another. As an application we reconstruct the f(T)
equivalent for quintessence, and we show that the deviation of the matter
overdensity evolution is strong for small scales and weak for large scales,
while it is negligible for large redshifts.
