Topological classification of affine operators on unitary and Euclidean spaces

We study affine operators on a unitary or Euclidean space U   up to topological conjugacy. An affine operator is a map f:U→Uf:U→U of the form f(x)=Ax+bf(x)=Ax+b, in which A:U→UA:U→U is a linear operator and b∈Ub∈U. Two affine operators f and g   are said to be topologically conjugate if g=h-1fhg=h-1fh for some homeomorphism h:U→Uh:U→U.If an affine operator f(x)=Ax+bf(x)=Ax+b has a fixed point, then f   is topologically conjugate to its linear part AA. The problem of classifying linear operators up to topological conjugacy was studied by Kuiper and Robbin [Topological classification of linear endomorphisms, Invent. Math. 19 (2) (1973) 83–106] and other authors.Let f:U→Uf:U→U be an affine operator without fixed point. We prove that f   is topologically conjugate to an affine operator g:U→Ug:U→U such that U is an orthogonal direct sum of g-invariant subspaces V and W,•the restriction g∣Vg∣V of g to V is an affine operator that in some orthonormal basis of V has the form(x1,x2,…,xn)↦(x1+1,x2,…,xn-1,εxn)(x1,x2,…,xn)↦(x1+1,x2,…,xn-1,εxn) uniquely determined by f  , where ε=1ε=1 if U   is a unitary space, ε=±1ε=±1 if U   is a Euclidean space, and n⩾2n⩾2 if ε=-1ε=-1, and•the restriction g∣Wg∣W of g to W is a linear operator that in some orthonormal basis of W is given by a nilpotent Jordan matrix uniquely determined by f, up to permutation of blocks.