Maps on matrices preserving local spectra

For a square matrix T and a nonzero vector eC(n), let sigma(T)(x) be the local spectrum of T at e. Characterization is obtained for surjective maps phi on M-n(C) satisfying sigma(phi(T)-phi(S))(e)sigma(T-S)(e) for all matrices T and S. The same description is obtained by supposing that sigma(T-S)(e)sigma(phi(T)-phi(S))(e) for all matrices T and S, without the surjectivity assumption on phi. Continuous maps from M-n(C) onto itself that preserve the local spectral radius distance at a nonzero fixed vector are also characterized.