question about the statement on Iooss's book

Question

I have started reading "Topics in Bifurcation theory" book by Gerard Iooss. I got a question on page 5, the book shortly says:

We consider nonlinear ordinary differential equation $$\frac{dz}{dt} = F(z),$$ where $z\in \mathbb{R}^n$, $t\in \mathbb{R}$ and $F \in C^k(\mathbb{R}^n; \mathbb{R}^n)$.

Assuming $F(0)=0$, we linearize this equation at $z=0$ and get $$\frac{dy}{dt} = Ly,$$ where $y \in \mathbb{R}^n$, and $L = DF(0)$.

Let $\sigma \subseteq \mathbb{C}$ be the spectrum of $L$. We split $\sigma$ into three disjoint parts $\sigma_-, \sigma_0$ and $\sigma_+$, where $$\sigma_- = \{ \lambda \in \sigma: \Re(\lambda) <0 \},$$ $$\sigma_0 = \{ \lambda \in \sigma: \Re(\lambda) =0 \},$$ $$\sigma_+ = \{ \lambda \in \sigma: \Re(\lambda) >0 \}.$$ Let $E_-, E_0, E_+$ be the $L$-invariant subspaces of $\mathbb{R}^n$ corresponding to the above splitting of $\sigma$. We have $$\mathbb{R}^n = E_- \cup E_0 \cup E_+.$$

My questions are:

How do we define those $E_-, E_0, E_+$ subspaces? What is the basis of each subspace?

Thanks in advance! Any hints are welcome!

Answer 1

If all eigenvalues were real, these spaces would simply be the direct sums of the corresponding generalized eigenspaces (or root spaces). That is, $$ E_-=\bigoplus_{\lambda\in\sigma_-}\left\{v\in\mathbb R^n:(L-\lambda I)^kv=0 \text{ for some } k\in\mathbb N\right\}, $$ $$ E_0=\bigoplus_{\lambda\in\sigma_0}\left\{v\in\mathbb R^n:(L-\lambda I)^kv=0 \text{ for some } k\in\mathbb N\right\}, $$ $$ E_+=\bigoplus_{\lambda\in\sigma_+}\left\{v\in\mathbb R^n:(L-\lambda I)^kv=0 \text{ for some } k\in\mathbb N\right\}. $$ In the general case one can show that $$ E_-=\mathbb R^n\cap\bigoplus_{\lambda\in\sigma_-}\left\{v\in\mathbb C^n:(L-\lambda I)^kv=0 \text{ for some } k\in\mathbb N\right\}, $$ $$ E_0=\mathbb R^n\cap\bigoplus_{\lambda\in\sigma_0}\left\{v\in\mathbb C^n:(L-\lambda I)^kv=0 \text{ for some } k\in\mathbb N\right\}, $$ $$ E_+=\mathbb R^n\cap\bigoplus_{\lambda\in\sigma_+}\left\{v\in\mathbb C^n:(L-\lambda I)^kv=0 \text{ for some } k\in\mathbb N\right\} $$ since nonreal eigenvalues come in pairs.

But really people usually (simply) note that $$ E_-=\left\{v\in\mathbb R^n:L^mv\to0 \text{ when } m\to+\infty\right\}, $$ $$ E_+=\left\{v\in\mathbb R^n:L^mv\to0 \text{ when } m\to-\infty\right\}, $$ while $E_0$ is the set of points whose orbit has no exponential speeds (future and past).