The embedding flows of $C^\infty$ hyperbolic diffeomorphisms

TL;DR

This paper provides a simplified proof that certain smooth hyperbolic diffeomorphisms can be embedded into smooth autonomous flows, clarifies the necessity of weakly nonresonant conditions, and characterizes when such embeddings exist in two dimensions.

Contribution

It offers a direct proof of embedding results, establishes the necessity of the weakly nonresonant condition, and characterizes embedding flows for two-dimensional hyperbolic diffeomorphisms.

Findings

Simplified proof of embedding hyperbolic diffeomorphisms into smooth flows.

Weakly nonresonant condition is necessary for smooth embedding.

Complete characterization of embedding flows in 2D case.

Abstract

In [{\it American J. Mathematics}, 124(2002), 107--127] we proved that for a germ of $C^\infty$ hyperbolic diffeomorphisms $F(x)=Ax+f(x)$ in $(\mathbb R^n,0)$, if $A$ has a real logarithm with its eigenvalues weakly nonresonant, then $F(x)$ can be embedded in a $C^\infty$ autonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field of $F(x)$ and the extension of the Floquet's theory to nonlinear $C^\infty$ periodic differential systems. In this paper we shall provide a simple and direct proof to this last result. Next we shall show that the weakly nonresonant condition in the last result on the real logarithm of $A$ is necessary for some $C^\infty$ diffeomorphisms $F(x)=Ax+f(x)$ to have $C^\infty$ embedding flows. Finally we shall prove that a germ of $C^\infty$ hyperbolic diffeomorphisms $F(x)=Ax+f(x)$ with $f(x)=O(|x|^2)$ in $(\mathbb R^2,0)$ has a $C^\infty$ embedding flow if and only if either $A$ has no negative eigenvalues or $A$ has two equal negative eigenvalues and it can be diagonalizable.