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

TL;DR

This paper revisits Arnold's theorem on intersection multiplicities under holomorphic maps, providing a new proof and extending it to local actions of finitely generated commutative groups, with implications for non-commutative cases.

Contribution

A new proof of Arnold's theorem using algebraic Noetherianity, generalizing it to broader group actions, and demonstrating limitations in non-commutative contexts.

Findings

New proof based on Noetherianity of algebras

Generalization to finitely generated commutative groups

Failure of analogous results for non-commutative groups

Abstract

V. I. Arnold proved in 1991 (published 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 to generalize the Arnold's theorem for local actions of arbitrary finitely generated commutative groups, both with discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails.