## Notes-09-14

4. Borel–Cantelli Lemma(s). Let $(X,\mathcal{X},\mu)$ be a probability space. Then

If $\sum_n \mu(A_n)<\infty$, then $\mu(x\in A_n \text{ infinitely often})=0$.

If $A_n$ are independent and $\sum_n \mu(A_n)=\infty$, then for $\mu$-a.e. $x$, $\frac{1}{\mu(A_1)+\cdots+\mu(A_n)}\cdot|\{1\le k\le n:x\in A_k\}|\to 1$.

The dynamical version often involves the orbits of points, instead of the static points. In particular, let $T$ be a measure-preserving map on $(X,\mathcal{X},\mu)$. Then

$\{A_n\}$ is said to be a Borel–Cantelli sequence with respect to $(T,\mu)$ if $\mu(T^n x\in A_n \text{ infinitely often})=1$;

$\{A_n\}$ is said to be a strong Borel–Cantelli sequence if $\frac{1}{\mu(A_1)+\cdots+\mu(A_n)}\cdot|\{1\le k\le n:T^k x\in A_k\}|\to 1$ for $\mu$-a.e. $x$.

3. Let $H(q,p,t)$ be a Hamiltonian function, $S(q,t)$ be the generating function in the sense that $\frac{\partial S}{\partial q_i}=p_i$. Then the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation

$H + \frac{\partial S}{\partial t}=0$.

Note that the total derivative $\frac{dS}{dt}=\sum_i\frac{\partial S}{\partial q_i}\dot q_i+\frac{\partial S}{\partial t}=\sum_i p_i\dot q_i-H=L$. Therefore, $S=\int L$ is the classical action function (up to an undetermined constant).

2. Let $\gamma_s(t)$ be a family of geodesic on a Riemannian manifold $M$. Then $J(t)=\frac{\partial }{\partial s}|_{s=0} \gamma_s(t)$ defines a vector field along $\gamma(t)=\gamma_0(t)$, which is called a Jacobi field. $J(t)$ describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic $\gamma$.

Alternatively, A vector field $J(t)$ along a geodesic $\gamma$ is said to be a Jacobi field, if it satisfies the Jacobi equation:

$\frac{D^2}{dt^2}J(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,$

where $D$ denotes the covariant derivative with respect to the Levi-Civita connection, and $R$ the Riemann curvature tensor on $M$.

1. Let $f:M\to M$ be an Anosov diffeomorphism, The Anosov shadowing lemma states that for any $\epsilon>0$, there exists $\delta\in(0,\epsilon)$ such that every $\delta$-psedo-orbit can be uniquely $\epsilon$-shadowed by a real orbit. Let $E\subset M$ be a finite $\frac{\delta}{2L(f)}$-dense subset of $M$, $\Sigma_E=\{\omega\in E^{\mathbb{Z}}:d(f\omega_n,\omega_{n+1})<\delta, \forall n\in\mathbb{Z}\}$. Then for each $\omega\in \Sigma_E$, let $\theta(\omega)\in M$ be the unique point whose orbit $\epsilon$-shadows $\omega$.

Our choice of constants also ensures that for every $x\in M$, there exists some $\omega\in\Sigma_E$ (may not be unique) such that $\theta(\omega)=x$: let $x_n\in E$ be (one of) the nearest point(s) to $f^nx$. In particular $d(x_n, f^nx)<\frac{\delta}{2L}$. Then $d(fx_n,x_{n+1})\le d(fx_n,f(f^nx))+d(f^{n+1}x,x_{n+1})<\frac{\delta}{2}+\frac{\delta}{2L}<\delta$. This induces a surjective map $\theta:\Sigma_E\to M$, which is clearly continuous.

GIven any two points $u, v\in\Sigma_E$ with $u_0=v_0$, we define a new point $w$ with $w_n=u_n$ for all $n\le 0$ and $w_n=v_n$ for all $n\ge 0$. Then it’s easy to see that $\theta(w)=[W^u(\theta u,\epsilon),W^s(\theta v,\epsilon)]$. That is, $\theta$ preserves the local produce structure. In particular, $R_\alpha=\theta([p_\alpha]_0)$ is a rectangle in $M$ for every $p_\alpha\in E$.

Let $x\in R_\alpha$ and $y\in W^s(x,R_\alpha)$. Then $x=\theta u$ for some $u\in \Sigma_E$ with $u_0=p_\alpha$; and $y=\theta v$ for some $v\in \Sigma_E$ with $v_0=p_\alpha$. Initially $v_n$ and $u_n$ may be different for some $n\ge 1$. However, we can always concatenate $[u,v]$, since $y\in W^s(x,\epsilon)$. So we assume $v_n=u_n$ for all $n\ge 0$. Then the trivial obervation $v_1=u_1=p_\beta$ can be translated to an interesting phenomenon: $fx\in R_\beta$ and $fy\in W^s(fx,R_\beta)$ and hence $fW^s(x,R_\alpha)\subset W^s(fx,R_\beta)$.

Note that $\{R_\beta:p_\beta\in E\}$ forms a covering of $M$ by closed rectangles and behaves like Markov partitions. However, $R_\beta$‘s are very likely to overlap. When $R_\alpha^o\cap R_\beta^o\neq\empty$, we introduc the following:

$R^1_{\alpha\beta}=\{x\in R_\alpha: W^u(x,R_\alpha)\cap R_\beta\neq\emptyset, W^s(x,R_\alpha)\cap R_\beta\neq\emptyset\}$;

$R^2_{\alpha\beta}=\{x\in R_\alpha: W^u(x,R_\alpha)\cap R_\beta\neq\emptyset, W^s(x,R_\alpha)\cap R_\beta=\emptyset\}$;

$R^3_{\alpha\beta}=\{x\in R_\alpha: W^u(x,R_\alpha)\cap R_\beta=\emptyset, W^s(x,R_\alpha)\cap R_\beta\neq\emptyset\}$;

$R^4_{\alpha\beta}=\{x\in R_\alpha: W^u(x,R_\alpha)\cap R_\beta=\emptyset, W^s(x,R_\alpha)\cap R_\beta=\emptyset\}$.

Note that each $R^j_{\alpha\beta}$ is a dynamical rectangle, since the defining relations above only rely on the local manifolds.

Then for an open dense subset of $x\in M$, let $R(x)=\bigcup\{R^j_{\alpha\beta}: x\in R_\alpha^o, R_\alpha^o\cap R_\beta^o\neq\empty, x\in R^j_{\alpha\beta}\}$. Note that $R(x)$ is also a dynamical rectangle, and form a partition (with repeated elements). Dropping the repeated ones, we have an refined partition with finitely many elements, say $\{R_1,\cdots, R_n\}$. Then check this inherits the Markov properties and forms a Markov partition.