# Hyperbolic dynamics.

1. Apr 27, 2008

### standardflop

Hello,
Given the three maps $x_{n+1}=Ax_n$ with
$$A_1=\begin{pmatrix} 1&-1\\1&1 \end{pmatrix}, A_2=\begin{pmatrix} 1/2&1/2\\-1&1 \end{pmatrix}, A_3=\begin{pmatrix} 3&2\\5/2&2 \end{pmatrix},$$
describe the dynamics, and say whether or not the dynamics is hyperbolic.

Finding eigenvalues and eigenvectors is relatively easy. And i know that the dynamics is hyperbolic if no eigenvalue lies on the unit circle (A1 is thus hyperbolic, and A2 is not). But is a system with eigenvalues {L1,L2} such that 0<L1<1<L2 hyperbolic (A3), or should both eigenvalues numerically be either greater or less than 1 ?

Also, does complex eigenvectors say anything about the general dynamics (cases: A1,A2) or is all that you can conclude that the origin is an unstable spiral because Re(Li)>0 ? I mean, in the case A3, the eigenvectors tells us the eigendirection of the stable and unstable subspaces, right?