possible dynamics on $\mathbb{R}^2$

Question

A linear map is non-hyperbolic if $|\lambda_i|=1$ for a least one eigenvalue $\lambda_i$.

Catalogue the possible dynamics of a non-hyperbolic linear map on $\mathbb{R}^2$

For something like this would you go through all the cases; i.e $|\lambda_1|=|\lambda_2|=1$ e.t.c?

Any hints on what all the cases are and mean would be great, thanks.

Answer 1

I would suggest the following. What happens if $|\lambda_1|=1$ and

1. $|\lambda_2|<1$ ?
2. $|\lambda_2|=1$ ?
3. $|\lambda_2|>1$ ?

Do you think we need to consider the cases where we swap $1$ and $2$ in the indices above?