## Tag Archives: Markov partition

### 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 $M$.