Week 10 Worksheet
- Prove that is measure-preserving and ergodic and mixing for Lebesgue measure on .
- Define by
for all . Take for granted that
is an orthonormal basis of . Prove is ergodic.
- Prove that irrational rotations on are not mixing.
- Fix a linear map from to . Prove that is continuous with respect to the norm if and only if is bounded in the sense that there is with
for all in .
- Fix a measure preserving map of a probability space . Suppose in is an eigenfunction of with eigenvalue .
- Prove that
and conclude .
- Prove that eigenfunctions with different eigenvalues are orthogonal.
- Prove that is a -invariant function.
- Prove, assuming is ergodic, that is constant almost everywhere.
- Prove, assuming is ergodic, that any other eigenfuction of is equal almost everywhere to a constant multiple of .