Tag Archives: chain recurrent

Some notes

Let (M^{2n},\omega)) be a symplectic manifold, S^{2p+q}\subset  M be a C^r submanifold such that when \omega is restricted to S, it has constant rank 2p. Then for each x\in S there exist a neighborhood U\ni x in M and C^{r-2} symplectic coordinates on (U,x_l,\cdots, x_{2n}), such that
S\cap U=\{p\in U: x_i(p)=0, i=p+q+1,\cdots,n,n+p+l,\cdots,2n\},

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

Suppose (X, \mathcal{X},\mu) is a probability space. The classical Borel–Cantelli lemmas state that
(1) if \{A_n:n\ge0\} is a sequence of measurable sets in X and
\sum_{n\ge0}\mu(A_n)<\infty, then
\mu(x\in X: x\in A_n\text{ for infinitely often times})=0.

(2) moreover if \{A_n:n\ge0\} is independent and \sum_{n\ge0}\mu(A_n)=\infty, then
\frac{S_n(x)}{E_n}\to 1 for \mu-a.e. x\in X.
where S_n(x) =\sum_{j=0}^n 1_{A_j}(x) and E_n=\sum_{j=0}^n \mu(A_j).

Suppose (X,\mu,T) is a measure-preserving transformation. If \{A_n\} is a sequence of sets such that \sum_n \mu(A_n)=\infty, is T^n(x)\in A_n for infinitely many times for
\mu-a.e. x\in X and, if so, is there a quantitative estimate of the asymptotic number
of entry times?

The shrinking target problem: A_n=B(x,r_n).

Let f:X\to X be a homeomorphism and \Omega_1(f)=\Omega(X,f) be the set of nonwandering points. Similarly we can define \Omega_2(f)=\Omega(\Omega_1(f),f) and inductively \Omega_{\alpha'}(f)=\Omega(\Omega_\alpha(f),f) for all \alpha\in\mathbb{N}. Let \Omega_\omega(f)=\bigcap_{\alpha\in\mathbb{N}}\Omega_\alpha(f). Once again \Omega_\omega(f) is a compact, invariant subset so we can restart our inductive process for all ordinals \alpha\in\mathbb{O}. If the process stabilizes at some ordinal \alpha, then we call it the center depth of (X,f) and \Omega_\alpha(f) the Birkhoff center of (X,f). There do exists examples that realize the countable ordinal \alpha\in\mathbb{O} (note there are uncountably many countable ordinals).

A point x\in X is said to be chain recurrent if for each \epsilon>0 there exists a nontrivial \epsilon-chain C_\epsilon initiated and terminated at x. Let \mathrm{CR}(f) be the set of chain recurrent points of (X,f). Unlike the nonwandering set, chain recurrent set always has depth 1:

Proposition. Let (X,f) be continuous. Then \mathrm{CR}(\mathrm{CR}(f),f)=\mathrm{CR}(f).
Proof. Let x\in \mathrm{CR}(f). Then for each \epsilon>0 there exists a \epsilon-chain C_\epsilon initiated and terminated at x. Let C_x=\limsup_{n\to\infty}C_{1/n}. If we can show that C_x\subset \mathrm{CR}(f), then for each \delta>0, pick m,n\ge1/\delta such that B(y,1/m)\subset B(fy,\delta) for all y\in X and C_{1/n}\subset B(\mathrm{CR}(f),1/m). Hence we \frac{1}{m}-perturb C_{1/n}=[x_k] to a new chain D_{1/n}=[y_k]\subset\mathrm{CR}(f). So d(fy_k,y_{k+1})\le d(fy_k,fx_k)+d(fx_k,x_{k+1})+d(x_{k+1},y_{k+1})<\delta+1/n+1/m<3\delta.

Now we are left to prove the claim. Let y\in C_x. For each \delta>0 we pick m,n\ge1/\delta such that B(y,1/m)\subset B(fy,\delta) for all y\in X and x_k\in C_{1/n}\cap B(y,1/m) for some k. We replace x_k by y to get a new chain (3\delta)
x=x_0\to\cdots\to x_{k-1}\to y\to x_{k+1}\to \cdots\to x_{K}=x, or equivalently
y\to x_{k+1}\to \cdots\to x_{K}=x=x_0\to\cdots\to x_{k-1}\to y. Since \delta can be arbitrarily small, we see that y is also chain recurrent. QED