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:


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.

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