4. Borel–Cantelli Lemma(s). Let be a probability space. Then

If , then .

If are independent and , then for -a.e. , .

The dynamical version often involves the orbits of points, instead of the static points. In particular, let be a measure-preserving map on . Then

– is said to be a Borel–Cantelli sequence with respect to if ;

– is said to be a strong Borel–Cantelli sequence if for -a.e. .

3. Let be a Hamiltonian function, be the generating function in the sense that . Then the *Hamilton–Jacobi equation* is a first-order, non-linear partial differential equation

.

Note that the total derivative . Therefore, is the classical action function (up to an undetermined constant).

2. Let be a family of geodesic on a Riemannian manifold . Then defines a vector field along , which is called a *Jacobi field*. describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic .

Alternatively, A vector field along a geodesic is said to be a *Jacobi field*, if it satisfies the *Jacobi equation*:

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

1. Let be an Anosov diffeomorphism, The Anosov shadowing lemma states that for any , there exists such that every -psedo-orbit can be uniquely -shadowed by a real orbit. Let be a finite -dense subset of , . Then for each , let be the unique point whose orbit -shadows .

Our choice of constants also ensures that for every , there exists some (may not be unique) such that : let be (one of) the nearest point(s) to . In particular . Then . This induces a surjective map , which is clearly continuous.

GIven any two points with , we define a new point with for all and for all . Then it’s easy to see that . That is, preserves the local produce structure. In particular, is a rectangle in for every .

Let and . Then for some with ; and for some with . Initially and may be different for some . However, we can always concatenate , since . So we assume for all . Then the trivial obervation can be translated to an interesting phenomenon: and and hence .

Note that forms a covering of by closed rectangles and behaves like Markov partitions. However, ‘s are very likely to overlap. When , we introduc the following:

– ;

– ;

– ;

– .

Note that each is a dynamical rectangle, since the defining relations above only rely on the local manifolds.

Then for an open dense subset of , let . Note that 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 . Then check this inherits the Markov properties and forms a Markov partition.