Locally finite polynomial endomorphisms

We study polynomial endomorphisms FF of CNCN which are locally finite in the following sense: the vector space generated by r∘Fnr∘Fn (n≥0n≥0) is finite dimensional for each r∈C[x1,…,xN]r∈C[x1,…,xN]. We show that such endomorphisms exhibit similar features to linear endomorphisms: they satisfy the Jacobian Conjecture, have vanishing polynomials, admit suitably defined minimal and characteristic polynomials, and the invertible ones admit a Dunford decomposition into “semisimple” and “unipotent” constituents. We also explain a relationship with linear recurrent sequences and derivations. Finally, we give particular attention to the special cases where FF is nilpotent and where N=2N=2.