## 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.

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 $p(m)$ is said to be the decay rate of a dynamical system $(X,\mu,T)$, if $|E(f\cdot g\circ T^m)-E(f)\cdot E(g)|\le p(m)\cdot\|f\|_{BV}\cdot\|g\|_{L^1}$ for all $g\in L^1(\mu)$ and all $f$ with bounded variation.

D. Kim; C. Gupta, M. Nicol andW. Ott: (summable decay of correlations implies the SBC property) assume the decay rates $p(m)$ satisfies $\sum_m p(m)<\infty$, then strong Borel-Cantelli property holds for any sequence of subsets $A_k$ with $\sum_k\mu(A_k)=\infty$.

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 $B_i$ with $\mu(B_i)\ge i^{-\gamma}$, for some $\gamma\in(0,1)$; (II) exponential decay
of correlations implies Borel–Cantelli for all the sequence of balls with $\mu(B_i)\ge i^{-1}$.

4. Borel-Cantelli Lemma. Let $(\Omega,P)$ be a probability space and $\{A_n\}$ be a sequence of events with $\sum_n P(A_n)<\infty$, then $P(A_n i.o.)=P(\limsup_{n\to\infty}A_n)=0$.

Proof. Consider $X=\sum_n I_{A_n}$ be the number of events that occur. Then the expectation $\mathbb{E}X<\infty$ and hence $X<\infty$, a.s.

Second Borel-Cantelli Lemma. Let $(\Omega,P)$ be a probability space and $\{A_n\}$ be a sequence of independent events with $\sum_n P(A_n)=\infty$, then $P(A_n i.o.)=P(\limsup_{n\to\infty}A_n)=1$.

Proof. Let $m. Then $P(\bigcup_{[k\ge m]}A_k)\ge 1-P(\bigcap_{[m,n]}A^c_k)=1-\prod_{[m,n]}(1-P(A_n))$
$\ge 1-\prod_{[m,n]}e^{-P(A_k)}=1-e^{-\sum_{[m,n]}P(A_k)}\to 1-e^{-\infty}=1$ as $n\to\infty$. So $P(\bigcup_{[k\ge m]}A_k)=1$ for all $m\ge1$ and hence $P(A_n i.o.)=P(\limsup_{n\to\infty}A_n)=1$.

In fact, the so called strong Borel-Cantelli property holds: $\displaystyle \frac{\sum_{1\le k\le n}I_{A_k}(x)}{\sum_{1\le k\le n}P(A_k)}\to 1$ for almost every point $x$.

Kolmogorov’s 0-1 Law. Let $\mathcal{T}=\bigcap_{n\ge1}\sigma(X_k:k\ge n)$ be the tail of $\sigma$-field (events of remote future). Let $X_1,\cdots,X_n,\cdots$ be independent and $A\in\mathcal{T}$ be a tail event. Then $P(A)\in\{0,1\}$.

Kolmogorov’s maximal inequality. Suppose $X_1,\cdots,X_n,\cdots$ are independent, $\mathbb{E}(X_i)=0$ and $\text{Var}(X_i)<\infty$. Then $P(\max_{1\le k \le n}|S_n|\ge r)\le r^{-2}\text{Var}(S_n)$.

Compare with Chebyshev inequality: $P(|S_n|\ge r)\le r^{-2}\text{Var}(S_n)$.

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3. Let $f:M\to M$ be a orientation-preserving diffeomorphism and $m$ be the volume measure induced by some volume form $\omega$. Let $D_xf:T_xM\to T_{fx}M$ be the tangent map between two normed space and $J(f,x)=\det(D_xf)$ be the Jacobian. We want to consider the Radon-Nikodym derivative of $f^\ast m$ with respect to $m$. To this end let $A\subset M$ be a measurable subset. Then

$f^\ast m(A)=m(f^{-1}A)=\int I_{f^{-1}A}(x)dm(x)$
$=\int_M I_{A}(fx)dm(x)=\int_M I_A(y)\cdot J(f^{-1},y)dm(y)$.
So $\displaystyle\phi(x)=\frac{df^\ast m}{dm}(x)=J(f^{-1},x)=\frac{1}{J(f,f^{-1}x)}$.

More generally we start with $d\mu=\phi\cdot dm$. Then $f^\ast\mu(A)=\mu(f^{-1}A)=\int I_{A}(fx)\phi(x)dm(x)$ $=\int I_A(y)\phi(f^{-1}y)\cdot J(f^{-1},y)dm(y)$. So $\frac{df^\ast\mu}{dm}(x)=\phi(f^{-1}x)\cdot J(f^{-1},x)$ and $\frac{df^\ast\mu}{d\mu}(x)=\frac{\phi(f^{-1}x)}{\phi(x)}\cdot J(f^{-1},x)$

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2. Let’s consider the ODE $\dot{x}=F(x)$. Suppose $\phi(t,x)$ solves the ODE with the initial $\phi(0,x)=x$. So $\dot{\phi}(t,x)=F(\phi(t,x))$. Taking differential w.r.t. $x$, we get a matrix equation: $D_x\dot{\phi}(t,x)=D_xF(\phi(t,x))\cdot D_x\phi(t,x)$. Then check that $\dot{J}(t,x)=\frac{d}{d t}(\det D_x\phi(t,x))=\mathrm{tr}(D_xF(\phi(t,x)))\cdot \det D_x\phi(t,x)$ $=\mathrm{div}F(\phi(t,x))\cdot J(t,x)$.
So the Jacobian is given by $J(t,x)=e^{\int_0^t \mathrm{div}F(\phi(s,x))dt}$. In particular $\dot{J}(0,x)=\mathrm{div}F(x)$.

For a vector field $X\in\mathfrak{X}(M)$, its divergence can be defined with respect to a given volume form $\omega$. That is, $\mathrm{div}X\cdot\omega=d(\imath_X\omega)=\mathfrak{L}_X\omega=\frac{d}{dt}|_{t=0}\phi^\ast_t\omega$.
So the vector field is divergence-free, $\mathrm{div}F=0$, if and only if the induced flow is volume-preserving.

Let $(M,g)$ be a Riemannian manifold, $\omega=\sqrt{\det g}dx^1\wedge\cdots\wedge dx^n$. Then $\mathrm{div}X\cdot\omega=d(\imath_X\omega)=(\sum\partial_i (V^i\sqrt{\det g}))dx^1\wedge\cdots\wedge dx^n=\frac{\sum\partial_i (V^i\sqrt{\det g})}{\sqrt{\det g}}\omega$. So $\mathrm{div}X=\frac{\sum\partial_i (V^i\sqrt{\det g})}{\sqrt{\det g}}$. $f:M\to\mathbb{R}$ be a smooth function and $\nabla\!f$ be the gradient vector field. Then the induced gradient flow is volume-preserving iff $\Delta\!f=\mathrm{div}(\nabla\!f)=0$, that is, $f$ is a harmonic function. According to maximum value principle, either $M$ is noncompact, or $f$ is a constant.

Let $(M,\omega)$ be a symplectic manifold, $H:M\to\mathbb{R}$ be a smooth function and $X_H$ be the symplectic gradient vector field. Then $\mathrm{div}(X_H)\omega^n=d(\imath_{X_H}\omega^n)=d(dH\wedge\omega^{n-1})=0$. 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: $\nabla\times(\nabla\phi)=\mathbf{0}$.

The divergence of the curl of any vector field $\mathbf{A}$ is always zero: $\nabla\cdot(\nabla\times\mathbf{A})=0$.

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1. Let $f\in\mathrm{PH}(M)$, $A_f(x)$ be the accessibility class containing the point $x$. There are some different levels of accessibility of $f$:

• topologically accessible: $\overline{A_f(x)}=M$ for some point $x\in M$,
• measure-theoretically accessible: $m(A)=0$ or $1$ for every measurable $su$-saturated set $A$,
• essentially accessible: $m(A_f(x))=1$ for some point $x\in M$ (the for a.e.),
• accessible: $A_f(x)=M$ for some point $x\in M$ (then for all).
• These are some formal definitions and need examples to distinguish them.