Take a dynamical map x
, with x=(p,q), which is
not integrable: there is no constant of motion F such that
Assume that the map is almost integrable, in the sense that
where x R(x) is integrable and &epsilon is infinitesimal.
Now you may want to define an approximate constant Fn of motion for your map, which is constant only up to a power n+1 in ε. That is, we want:
An iterative process leads us to
with a relation of the kind
Then the series (3) does not converge uniformly. Indeed if it was the case it would define a constant of motion and it would mean that M is an integrable map, which has been discarded.
On the other hand, if you fix n and make &epsilon tend to 0 then relation (3) holds so we have indeed an approximate constant of motion.
Practically it means that for a given &epsilon there is an optimal n for which the series (4) is an approximate constant of motion.