The author shows that if \(U\) is a wandering domain of a meromorphic function \(f\) such that there exists a point \(z_0 \in U\) and a neighborhood \(V \subset U\) of \(z_0\) where one of the following properties holds for every \(w \in V\):
(a) \(d_{U_n}(f^n (w), f^n(z_0)) \to 0\) (\(V\) is contracting relative to \(z_0\));
(b) \(d_{U_n}(f^n(w), f^(z_0))\) decreases to a limit \(c(z_0, w) > 0\) without ever reaching it, except for a discrete (in \(V\)) set of points for which \(f^k(w) = f^k(z_0)\) for some \(k \in \mathbb{N}\) (\(V\) is semicontracting relative to \(z_0\));
(c) There exists \(N \in \mathbb{N}\) (uniform over compact subsets of \(V\)) such that \(d_{U_n}(f^n(w), f^n(z_0)) = c(z_0, w) > 0\) for \(n \geq \mathbb{N}\) (\(V\) is locally eventually isometric relative to \(z_0\));
then the same property holds for every \(w \in U\).
As an application of the above result, the author constructs an example of a meromorphic function, by using quasiconformal surgery, with a semi-contracting infinitely connected wandering domain. The example is stated as follows: there exists a meromorphic function \(f\) with an infinitely connected wandering domain \(V\) and a nonempty open subset \(V' \subset V\) such that, for \(z_0 \in V'\), \(V\) is semicontracting relative to \(z_0\).