Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

TL;DR

This paper offers a new, simplified proof that hyperbolic diffeomorphisms with weakly nonresonant eigenvalues can be embedded in smooth flows, and characterizes embeddability conditions including the impact of resonances.

Contribution

It introduces a novel proof using Jordan-Chevalley decomposition, characterizes embeddability of diffeomorphisms, and proves a conjecture regarding weak resonances.

Findings

New proof of embedding theorem using algebraic group decomposition

Characterization of embeddable diffeomorphisms with weak resonances

A criterion showing when weak resonances prevent embeddability

Abstract

In [Xiang Zhang, The embedding flows of $C^{\infty}$ hyperbolic diffeomorphisms, J. Differential Equations 250 (2011), no. 5, 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new, simpler and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator. We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterium showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterium is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.