I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic theorem (BET), which is why I'd stated VNMET in the first place. But I'm pretty sure that VNMET came first, so I doubt it was originally proven with that in mind.

Question 1. How did the theorem (or conjecture) arise in the first place? E.g. was it intended as as mere stepping-stone to BET?

The only applications I've seen (and can find) of ergodic theory to other branches of math (or physics) use BET.

Question 2. What are some applications of VNMET? I'm particularly interested in applications to other branches of mathematics; so I'm looking for something other than the "application" of it to prove BET.

Edit. What I'm really trying to glean with the above questions is an answer to the following question:

Question 3. Is VNMET important/useful beyond proving BET?


von Neumann long argued that for physics, his result suffices (see, e.g., Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 263–266,). There is not only truth to that but also to the fact that his result suffices for some of the mathematical applications. Moreover, as von Neumann emphasized [in the above], there is one aspect of his result that is stronger than Birkhoff’s. If one defines $$Av(n,L) (\omega; f) = \frac{1}{n} \sum_{j=L}^{n+L-1} f(T_j(\omega))$$ then as $n \rightarrow\infty$, in $L^2$, $Av(n,L) ( · ; f)$ converges uniformly in $L$ (as can be seen by looking at either the von Neumann or Hopf proofs), but the pointwise convergence need not be uniform in $L$.

  • $\begingroup$ @Barry This seems like a great answer and I will probably accept it. Before I do, I must understand it better. In particular, I need to read the paper you cited in more depth. $\endgroup$ – Quinn Culver Aug 22 '11 at 1:02

A nice, not too well-known application of the von Neumann ergodic theorem, is to the classification of (surjective) isometries of a Hilbert space. View the Hilbert space $H$ as a metric space. By the Mazur-Ulam theorem (easy in this case, as $H$ is strictly convex), every isometry $\alpha$ is affine, i.e. $\alpha(x)=Ux+b$ for every $x\in H$, where $U$ is a unitary operator on $H$ (the linear part of $\alpha$) and $b\in H$ (the translation part of $\alpha$). The following equivalence is classical:

i) $\alpha$ has a fixed point;

ii) $b$ is in the image of $U-1$;

iii) the sequence $(\|\alpha^n(0)\|)_{n\geq 0}$ is bounded.

The following equivence is less classical:

i') $\alpha$ almost has a fixed point;

ii') $b$ is in the closure of the image of $U-1$;

iii') $\|\alpha^n(0)\|=o(n)$.

To prove the latter equivalence, observe that $\alpha^n(0)=(1 + U + ... + U^{n-1})b$; remembering that the kernel of $U-1$ is the orthogonal of the image of $U-1$ (as $U-1$ is a normal operator), we see that the von Neumann ergodic theorem is responsible for the equivalence $(ii')\Leftrightarrow (iii')$.

  • $\begingroup$ @Alain Thanks for the nice answer. Where can I find the "less classical" equivalence? $\endgroup$ – Quinn Culver Aug 21 '11 at 12:55
  • 2
    $\begingroup$ @ Quinn: See e.g. my notes "Isometric actions on Hilbert spaces and amenability", available on www2.unine.ch/math/page-6197.html#cid114886 $\endgroup$ – Alain Valette Aug 21 '11 at 21:00

I believe that Hillel Furstenberg uses the von Neumann ergodic theorem quite frequently in his work on recurrence, which has applications to number theory. For example, in section 3 of the article Poincaré recurrence and number theory he uses Weyl's criterion and the von Neumann ergodic theorem to prove the following result: if $T$ is a measure-preserving transformation of a probability space, $p$ is a polynomial with integer coefficients and no constant term, and $A$ is a positive-measure subset of the probability space in question, then there are infinitely many natural numbers $t$ such that $T^{-p(t)}A \cap A$ has postive measure. A corollary of this result is that if $X$ is a subset of the integers with positive density and $p$ is an integer polynomial with no constant term, then the equation $x-y=p(t)$ can be solved for $x,y \in X$ and $t$ a positive integer. The von Neumann ergodic theorem is also used in ergodic proofs of Roth's theorem (see for example the exposition by Á. Magyar). There are probably more examples in the book Recurrence in Ergodic Theory and Combinatorial Number Theory.


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