$$$$

Put $X=[0,1{)}^2$. Define $T:X\to X$ by $$T(x,y)=(2x+ymod1,x+ymod1)$$ for all $x,y\in [0,1)$.

1. What is the orbit of $(\frac{1}{2},\frac{1}{2})$?
2. What is the orbit of $(\frac{1}{3},\frac{2}{3})$?
3. What is the connection between $T$ and the matrix $A=[\begin{array}{cc}2& 1\\ 1& 1\end{array}]$?
4. Fix a rectangle $R\subset {\mathbb{R}}^2$. 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 $\mathcal{B}$ 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 $(X,\mathcal{B})$ with respect to which $T$ is measure-preserving.

7. What are the eigenvalues and eigenvectors of $A$?
8. Let $V$ be any subspace of ${\mathbb{R}}^2$ spanned by an eigenvector of $A$. Is $$\{(xmod1,ymod1):(x,y)\in V\}$$ dense in $[0,1{)}^2$?
9. Is there a point in $V$ whose $T$ orbit is dense in $[0,1)$?