Local dynamics of intersections: V. I. Arnold’s theorem revisited

V. I. Arnold proved in 1993 that the intersection multiplicity between two germs of analytic subvarieties at a fixed point of a holomorphic invertible self-map remains bounded when one of the germs is dragged by iterations of the self-map. The proof is based on the Skolem-Mahler-Lech theorem on zeros in recurrent sequences.

We give a different proof, based on the Noetherianity of certain algebras, which allows one to generalize Arnold’s theorem for local actions of arbitrary finitely generated commutative groups, with both discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails.