## Tag Archives: Borel-Cantelli

### 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 $M$.

### Some notations

7. Let $f$ be an Anosov diffeomorphism and $g\in\mathcal{U}(f)$ be close enough, which leads to a Holder continuous conjugate $h_g:M\to M$ with $g\circ h_g=h_g\circ f$. Ruelle found an explicit formula of $h_g$.

Let $f,g:M\to M$ be two homeomorphisms, $d(f,g)=\sup_M d(fx,gx)$, and $\mathcal{U}(f,\epsilon)=\{g \text{ homeo and }d(f,g)<\epsilon\}$. Let $g\in \mathcal{U}(f,\epsilon)$. Then the map $X_g:x\in M \mapsto \exp^{-1}_{fx}(gx)\in T_{fx}M$ gives a shifted-vector field on $M$, which induces a diffeomorhism $\mathcal{U}(f,\epsilon)\to \mathcal{X}(0_f,\epsilon), g\mapsto X_g$.
Let $f$ be a $C^r$ diffeomprhism. Then $\mathcal{X}^r(0_f,\epsilon)\to \mathcal{U}^r(f,\epsilon), g\mapsto X_g$ induces the local Banach structure and turns $\mathrm{Diff}^r(M)$ into a Banach manifold.

Let $X_g\circ f^{-1}=X_g^s+X_g^u$ be the decomposition of the correction $X_g\circ f^{-1}$ with respect to the hyperbolic splitting $TM= E_g^s\oplus E_g^u$. Then the derivative of $g\mapsto h_g$ in the direction of $X_g$ is given by the vector field $\displaystyle \sum_{n\ge 0}Dg^n X^s_g-\sum_{n\ge1}Dg^{-n}X^u_g$.

6. Let $M$ be a compact orientable surface of genus $g\ge1$, $s\ge1$ and let $\Sigma=\{p_1,\cdots,p_s\}$ be a subset of $M$. Let $\kappa= (\kappa_1,\cdots,\kappa_s)$ be a $s$-tuple of positive integers with $\sum (\kappa_i-1) =2g-2$.

A translation structure on $(M,\Sigma)$ of type $\kappa$ is an atlas on $M\backslash\Sigma$
for which the coordinate changes are translations, and such that each singularity $p_i$
has a neighborhood which is isomorphic to the $\kappa_i$-fold covering of a neighborhood
of $0$ in $\mathbb{R}^2\backslash\{0\}$.

The Teichmüller space $Q_{g,\kappa}= Q(M,\Sigma,\kappa)$ is the set of such structures modulo isotopy relative to $\Sigma$. It has a canonical structure of manifold.