  A relevant perturbation of the entanglement entropy of a sphere is examined
holographically near the UV fixed point. Varying the conformal dimension of the
relevant operator, we obtain three different sectors: 1) the entanglement
entropy is stationary and the perturbative expansion is well-defined with
respect to the relevant coupling, 2) the entropy is stationary, but the
perturbation fails, 3) the entropy is neither stationary nor perturbative. We
compare our holographic results with the numerical calculation for a free
massive scalar field in three-dimensions, and find a qualitative agreement
between them. We speculate that these statements hold for any relevant
perturbation in any quantum field theory invariant under the Poincare symmetry.
