Let be a symplectic manifold, be a submanifold such that when is restricted to , it has constant rank . Then for each there exist a neighborhood in and symplectic coordinates on , such that

,

i. e., there exist symplectic coordinates on that give as a vector subspace.

Suppose is a probability space. The classical Borelâ€“Cantelli lemmas state that

(1) if is a sequence of measurable sets in and

, then

.

(2) moreover if is independent and , then

for a.e. .

where and .

Suppose is a measure-preserving transformation. If is a sequence of sets such that , is for infinitely many times for

a.e. and, if so, is there a quantitative estimate of the asymptotic number

of entry times?

The shrinking target problem: .

Let be a homeomorphism and be the set of nonwandering points. Similarly we can define and inductively for all . Let . Once again is a compact, invariant subset so we can restart our inductive process for all ordinals . If the process stabilizes at some ordinal , then we call it the center depth of and the Birkhoff center of . There do exists examples that realize the countable ordinal (note there are uncountably many countable ordinals).

A point is said to be chain recurrent if for each there exists a nontrivial chain initiated and terminated at . Let be the set of chain recurrent points of . Unlike the nonwandering set, chain recurrent set always has depth 1:

Proposition. Let be continuous. Then .

Proof. Let . Then for each there exists a chain initiated and terminated at . Let . If we can show that , then for each , pick such that for all and . Hence we perturb to a new chain . So .

Now we are left to prove the claim. Let . For each we pick such that for all and for some . We replace by to get a new chain ()

, or equivalently

. Since can be arbitrarily small, we see that is also chain recurrent. QED