7. Let be an Anosov diffeomorphism and be close enough, which leads to a Holder continuous conjugate with . Ruelle found an explicit formula of .

Let be two homeomorphisms, , and . Let . Then the map gives a shifted-vector field on , which induces a diffeomorhism .

Let be a diffeomprhism. Then induces the local Banach structure and turns into a Banach manifold.

Let be the decomposition of the correction with respect to the hyperbolic splitting . Then the derivative of in the direction of is given by the vector field .

6. Let be a compact orientable surface of genus , and let be a subset of . Let be a -tuple of positive integers with .

A translation structure on of type is an atlas on

for which the coordinate changes are translations, and such that each singularity

has a neighborhood which is isomorphic to the -fold covering of a neighborhood

of in .

The Teichmüller space is the set of such structures modulo isotopy relative to . It has a canonical structure of manifold.

5. Dynamical Borel–Cantelli lemmas. Chernov and Kleinbock established the SBC property for certain families of cylinders in the setting of topological Markov chains and for certain classes of dynamically defined rectangles in the setting of Anosov diffeomorphisms preserving Gibbs measures. Dolgopyat has related BC results for sequences of balls in uniformly partially hyperbolic systems preserving a measure equivalent to Lebesgue which

have exponential decay of correlations with respect to Hölder observables.

A sequence of real numbers is said to be the decay rate of a dynamical system , if for all and all with bounded variation.

D. Kim; C. Gupta, M. Nicol andW. Ott: (summable decay of correlations implies the SBC property) assume the decay rates satisfies , then strong Borel-Cantelli property holds for any sequence of subsets with .

Haydn, Nicol, Persson and Vaienti: (I) under certain assumptions on the measure, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies Borel–Cantelli for all sequence of balls with , for some ; (II) exponential decay

of correlations implies Borel–Cantelli for all the sequence of balls with .

4. **Borel-Cantelli Lemma.** Let be a probability space and be a sequence of events with , then .

Proof. Consider be the number of events that occur. Then the expectation and hence , a.s.

**Second Borel-Cantelli Lemma.** Let be a probability space and be a sequence of *independent* events with , then .

Proof. Let . Then

as . So for all and hence .

In fact, the so called *strong Borel-Cantelli* property holds: for almost every point .

**Kolmogorov’s 0-1 Law.** Let be the tail of -field (events of remote future). Let be independent and be a tail event. Then .

**Kolmogorov’s maximal inequality.** Suppose are independent, and . Then .

Compare with Chebyshev inequality: .

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3. Let be a orientation-preserving diffeomorphism and be the volume measure induced by some volume form . Let be the tangent map between two normed space and be the Jacobian. We want to consider the Radon-Nikodym derivative of with respect to . To this end let be a measurable subset. Then

.

So .

More generally we start with . Then . So and

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2. Let’s consider the ODE . Suppose solves the ODE with the initial . So . Taking differential w.r.t. , we get a matrix equation: . Then check that .

So the Jacobian is given by . In particular .

For a vector field , its divergence can be defined with respect to a given volume form . That is, .

So the vector field is divergence-free, , if and only if the induced flow is volume-preserving.

Let be a Riemannian manifold, . Then . So . be a smooth function and be the gradient vector field. Then the induced gradient flow is volume-preserving iff , that is, is a harmonic function. According to maximum value principle, either is noncompact, or is a constant.

Let be a symplectic manifold, be a smooth function and be the symplectic gradient vector field. Then . That is, (time-independent) Hamiltonian flow is always volume-preserving (the time-dependent version is also true and preserves the symplectic form).

The curl of the gradient is always the zero vector: .

The divergence of the curl of any vector field is always zero: .

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1. Let , be the accessibility class containing the point . There are some different levels of accessibility of :

These are some formal definitions and need examples to distinguish them.