5
$\begingroup$

Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville measure.

A classical (but still wonderful) remark due to Koopman is that given a measure preserving map $\phi: M \rightarrow M$, the operator $$ U_\phi : L_2(M,|\omega^n|) \longrightarrow L_2(M,|\omega^n|) $$ defined by $f \mapsto f \circ \phi$ is unitary. The operator $U_\phi$ is sometimes called the Koopman operator of the map $\phi$. Since symplectomorphisms are measure preserving, it seems natural to ask the following

Vague question. Is there anything at all particular about Koopman operators of symplectomorphisms?

$\endgroup$
  • 1
    $\begingroup$ Assuming that by "uniform closure", you mean "norm closure", then operators that you describe form a closed discrete set (any tow of them have distance 2). $\endgroup$ – André Henriques Jun 27 '14 at 12:39
  • $\begingroup$ Of course !! It was a dumb way to try to make precise a vague question. I'm editing the question to leave it vague. ;-) $\endgroup$ – alvarezpaiva Jun 27 '14 at 12:48
  • $\begingroup$ What is the motivation for your question? $\endgroup$ – André Henriques Jun 27 '14 at 15:32
  • $\begingroup$ @AndréHenriques: I'm trying to see if there is any symplectic version of ergodic theory, one that uses that the maps are not just measure preserving. $\endgroup$ – alvarezpaiva Jun 27 '14 at 17:13
  • $\begingroup$ Which results/concepts in ergodic theory do you expect to be affected by the fact that the transformation is a symplectomorphism? $\endgroup$ – André Henriques Jun 27 '14 at 19:05
2
$\begingroup$

The Koopman operator is bracket preserving with respect to the canonical Poisson bracket on $C^{\infty}(M)$. Formally, this implies the dual operator (the Frobenius-Perron operator) is a Poisson automorphism with respect to the Lie-Poisson structure on $\bigwedge^{2n}(M)$.

$\endgroup$
  • $\begingroup$ Thanks. This is the usual "functional" description of symplectic maps, but how to "marry" this with ergodic theory? Perhaps we have to change the space of square integrable functions for a Sobolev space and have the Koopman operator act on it instead. $\endgroup$ – alvarezpaiva Aug 5 '14 at 10:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.