Locally finite polynomial endomorphisms

We study polynomial endomorphisms F of C^N which are locally finite in the following sense: the vector space generated by r°Fⁿ (n≥0) is finite dimensional for each r∈Cx₁,…,x_N]. 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 F is nilpotent and where N=2.