## 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