Proper holomorphic self-maps of plane regions

If ω is a connected open set in C, whose connectivity is finite hut larger than 2, then every proper holomorphic map from R to Cl is one-to-one. On the orher hand , for every positive integer m there are regions ω ⊂C, of infinite connectiviry, which admit proper holomorphic self-maps of mulitiplicity m. ∗ ∗This research was partially supported by an NSF Grant and by the E W. Vilas 2ust Estate.