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.

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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.

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