Configuration spaces of C and CP¹: some analytic properties

We study holomorphic self-maps of non-ordered n-point configuration spaces C^n(X), where X is either affine or projective complex line. The complex Lie group Aut(X) acts diagonally on C^n(X). We prove that for n>4 every endomorphism F of C^n(X) either is tame meaning that it is of the form F(Q)=T(Q)Q for a certain morphism T of C^n(X) to Aut(X) or carries the whole C^n(X) into one Aut(X) orbit Aut(X)Q_0 in C^n(X). The first option happens if and only if the image of the induced endomorphism F_* of the fundamental group of C^n(X) (which is the braid group of X) is a non-cyclic group; otherwise F is of the second type. We also prove that for n>(dim(Aut(X))+1 a morphism F of C^n(X) to any C^k(X) always admits an n point subset Q of X whose intersection with its image F(Q) is non-empty. Finally, we give a complete description of unbranched k-coverings of C^n(x) for k<2n+1.