Week 8 Tutorial Solutions
- By multiplying the matrix with and then reducing the entries mod one, one gets .
- The matrix has determinant so areas of polygons are not changed when mapped by . Thus and have the same area.
- Let be the collection of all Cartesian rectangles in together with . This is a π-system so it suffices to check that and have the same area. But and have the same area, and reducing modulo one does not cause any overlap, so has the same area as .
- Set
which is invariant because cannot change the cardinalty of
and
have the same cardinality for all .
- The eigenvalues of are the roots of its characteristic polynomial . These are
with
the respective eigenvalues.
- As each eigenspace wraps around the unit square, its consecutive intersections with the bottom of the square correspond to the orbit of an irration rotation. As such orbits are dense in the unit interval, and as the slope does not change, we get that the eigenspace is dense in .
- We will be able to prove the existence of points with dense orbit after developing some more theory.