So, what’s ergodic theory all about?
The idea is that we are given a system together with some operation on it. For instance, could be a homeomorphism of a topological space, i.e. a discrete dynamical system. We are interested in studying the iterates of this process. In many case, averaging over the iterates of this process yields in the limit something that is actually invariant under this process.
For instance, suppose is a measure-preserving transformation of a measure space (which means if is measurable, then so is and ). How might one arise? Well, suppose is a compact symplectic manifold, a Hamiltonian vector field, and the volume form. Then the flows of leave invariant the volume form , so any such diffeomorphism is a measure-preserving transformation of the measure induced by the volume form. Anyway, back to the general story. Then the action of on a function is given by
The Birkhoff ergodic theorem states that the averages
converge a.e. to a function invariant under , provided . In many interesting cases, the invariant limit will actually be constant a.e. For instance, this is guaranteed if the transformation is ergodic, i.e. has no nontrivial invariant subsets. There are many spectacular applications to number theory of this result, e.g. the existence of Khintchine’s constant. Cf. also this post of Harrison Brown.
Today, I’d like to talk about a simpler result in functional analysis. The idea is to replace with , and a.e. convergence with convergence. It turns out that the reason for this is to work more generally with operators in Hilbert spaces.
This is the von Neumann ergodic theorem:
Theorem 1Let be a unitary operator on a Hilbert space . Consider the closed subspace of vectors of left invariant by , and let be the orthogonal projection. Then for all ,
The proof of this result is actually extremely simple: decompose into two parts, and verify the claim for each!
One of these components is . The other is the closed span of . I claim that . It will be easy to verify the theorem for or . Suppose now . Then for all :
which shows that . Conversely, if , then for any
by assumption, proving .
Now we will prove the equality for both subspaces .
First, for vectors in , the result is seen as follows. The limit on the left is zero by a telescoping type argument. Also, , because and are orthogonal.
Now suppose . Then the sum on the left is always since , and by assumption as well.