Invariant sets through resonant normal form for infinite dimensional holomorphic vector fields

TL;DR

This paper investigates invariant sets for infinite dimensional holomorphic vector fields on sequence spaces, establishing conditions for the existence of invariant submanifolds where dynamics simplify to linear behavior.

Contribution

It introduces a method to construct invariant submanifolds in infinite dimensional settings using resonant normal form techniques under Diophantine-like conditions.

Findings

Existence of analytic invariant submanifolds passing through the fixed point.

Restricted dynamics are analytically conjugate to linear dynamics.

Conditions under which invariant sets exist in infinite dimensions.

Abstract

In this paper, we study infinite dimensional holomorphic vector fields on sequence spaces, having a fixed point at $0$. Under suitable hypotheses we prove the existence of analytic invariant submanifolds passing through the fixed point. The restricted dynamics is analytically conjugate to the linear one under some Diophantine-like condition.