Configuration spaces of C and CP¹: some analytic properties

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We study holomorphic self-maps of non-ordered n-point configuration spaces Cⁿ(X), where X is either affine or projective complex line. The complex Lie group Aut(X) acts diagonally on Cⁿ(X). We prove that for n>4 every endomorphism F of Cⁿ(X) either is tame meaning that it is of the form F(Q)=T(Q)Q for a certain morphism T of Cⁿ(X) to Aut(X) or carries the whole Cⁿ(X) into one Aut(X) orbit Aut(X)Q₀ in Cⁿ(X). The first option happens if and only if the image of the induced endomorphism F* of the fundamental group of Cⁿ(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 Cⁿ(X) to any Cᵏ(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 Cⁿ(x) for k<2n+1.