Dynamics of a linear map 
Let $F : \mathbb{R^2} → \mathbb{R^2}$ be any map and, given a point
  $(x_0, y_0)$ in $\mathbb{R^2}$ define $x_n$ and $y_n$ by $(x_{n+1}, y_{n+1}) = F(x_n, y_n)$. We study the dynamics of the map $F$ by
  studying the limiting behaviour of the sequence $(x_n, y_n)$ as $n→∞$
  for different choices of the starting point $(x_0, y_0)$. What is the
  limiting behaviour of $(x_n, y_n)$ when $F$ is expressed by
$$\bigg(\begin{matrix} 0~2\\ 1~1\\ \end{matrix}\bigg)$$

I thought I'll try to do this using Cayley-Hamilton theorem, but the hint suggests:

Find the eigenvalues of the matrix. The lines $y = x$ and $x + 2y = 0$ should figure prominently in your solution.

I can see, that the lines in the hint are just eigenvectors, and if we take points that are eigenvectors, then the dynamics are clear. But what about other points? Does the hint somehow helps to to find dynamics for those points or is Cayley-Hamilton theorem the way to go?
 A: Other points are linear combinations of eigenvector, hence by linearity their behviour can be predicted componentwise with respect to this decomposition.
A: Suppose $F$ has two linearly independent eigenvectors, namely $v_1$ and $v_2$, with eigenvalues res. $\lambda_1$ and $\lambda_2$. representation of $F$ in $\{v_1,v_2\}$ basis is :
$$
        \begin{pmatrix}
        \lambda_1 & 0  \\
        0 & \lambda_2  \\
        \end{pmatrix}
$$
For any vector $v = c_1 v_1 + c_2 v_2$ we have $T(v) = c_1\lambda_1 v_1 + c_2\lambda_2 v_2$ . doing this $n$ times gives $T^n(v) = c_1\lambda_1^n v_1 + c_2\lambda_2^n v_2$. if we set $c_2 = 0$, we have $T^n(v) = c_1\lambda_1^2 v_1$, so $T$ maps $\langle v_1\rangle$  into itself. if $|\lambda_1| < 1$, then future of all points in $\langle v_1\rangle$ goes to $0$ as $n \to \infty$.if $|\lambda_1| > 1$, then the future of points is unbounded. same holds for $\langle v_2\rangle$ and $\lambda_2$.
For analyzing orbits of general vectors,  you should think of  three cases: $|\lambda_1| , |\lambda_2| <1$, $|\lambda_1| , |\lambda_2| > 1$ and $|\lambda_1| < 1 , |\lambda_2| > 1$. for each case you should ask questions like: which orbits are bounded? which orbits converge to a limit? is there any asymptotic behavior?
