  If f: C -> P^n is a holomorphic curve of hyper-order less than one for which
2n + 1 hyperplanes in general position have forward invariant preimages with
respect to the translation t(z)=z+c, then f is periodic with period c. This
result, which can be described as a difference analogue of M. Green's
Picard-type theorem for holomorphic curves, follows from a more general result
presented in this paper. The proof relies on a new version of Cartan's second
main theorem for the Casorati determinant and an extended version of the
difference analogue of the lemma on the logarithmic derivatives, both of which
are proved here. Finally, an application to the uniqueness theory of
meromorphic functions is given, and the sharpness of the obtained results is
demonstrated by examples.
