Problem Sheet O

This Problem Sheet is based on Lectures 29 and 30.

• Problem 1 shows that our "new" definition of hyperbolicity is consistent with the old definition from Lecture 8.
• Problems 2 and 3 give two equivalent ways of defining hyperbolicity.
• If $L$ is hyperbolic then with respect to the hyperbolic splitting $E^s \oplus E^u$, $L$ has a block diagonal matrix form:
  $$
  L =
  \begin{pmatrix}
  L_{ss} & 0 \\
  0 & L_{uu}
  \end{pmatrix},
  $$
  where (with respect to an adapted norm)
  $$
  \max \big\{ \|L_{ss}\|^{\operatorname{op}}, \|L_{uu}\|^{\operatorname{op}} \big\} < 1.
  $$
  Problem 4 generalises this to operators which have a (upper/lower) triangular representation with respect to a splitting.
• Problem 5 isn't strictly related to this course, but it's good to know, and we will use it later on. You may have seen this result before in your linear algebra/functional analysis course.
• Problem 6 shows that all contracting reversible linear dynamical systems on a one-dimensional real vector space are conjugate, but that it is never possible to choose the conjugacy in such a way that it is bi-Lipschitz. We will see more instances of this phenomenon later on in the course.