\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\P}{\mathcal{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\d}{\mathsf{d}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \newcommand{\B}{\mathscr{B}} \newcommand{\fou}{\mathcal{F}} \]

Put $X = [0,1)^2$. Define $T : X \to X$ by \[ T(x,y) = (2x + y \bmod 1, x+y \bmod 1) \] for all $x,y \in [0,1)$.

1. What is the orbit of $(\tfrac{1}{2},\tfrac{1}{2})$?
2. What is the orbit of $(\tfrac{1}{3},\tfrac{2}{3})$?
3. What is the connection between $T$ and the matrix $A = [\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}]$?
4. Fix a rectangle $R \subset \R^2$. What is the area of the polygon $A^{-1}(R)$?

Take for granted that there is a measure $\lambda$ on $X$ on the σ-algebra $\mathscr{B}$ generated by the rectangles in $[0,1)^2$ that assigns each rectangle $S \subset [0,1)^2$ its Euclidean area.

5. Prove that with respect to $\lambda$ the map $T$ is measure-preserving.
6. Using 1. or 2. write down another measure on $(X,\B)$ with respect to which $T$ is measure-preserving.

7. What are the eigenvalues and eigenvectors of $A$?
8. Let $V$ be any subspace of $\R^2$ spanned by an eigenvector of $A$. Is \[ \{ (x \bmod 1, y \bmod 1) : (x,y) \in V \} \] dense in $[0,1)^2$?
9. Is there a point in $V$ whose $T$ orbit is dense in $[0,1)$?