## New notes

4. Ledrappier-Young’s criterion for measures with Absolutely Continuous Conditional measures on Unstable manifolds. Let $f:M\to M$ be a $C^2$ diffeomorphism and $\mu$ be an invariant probability measure. For $\mu$-a.e. $x$, if there exist negative Lyapunov exponent(s) at $x$, then the set of points with exponentially approximating future of $x$ is a $C^2$-submanifold, $W^u(x)$. Then $\mu$ is said to have ACCU if for each measurable partition $\xi$ with $\xi(x)\subset W^u(x)$ and contains an unstable plaque for $\mu$-a.e. $x$, the conditional measure $\mu_{\xi(x)}\ll m_{W^u(x)}$ for $\mu$-a.e. $x$.

Theorem. $\mu$ has ACCU if and only if $h_\mu(f)=\Lambda^+(\mu)$, where $\Lambda^+(\mu)=\int\sum\lambda^+_i(x)d\mu(x)$.

Definition. Such a measure is called the SRB measure.

Moreover, the density $\frac{d\mu_{\xi(x)}}{dm_{W^u(x)}}$ is strictly positive and $C^1$ on $\xi(x)$ for $\mu$-a.e. $x$.

3. Exponential tail bound for coupling lemma. Suppose that for each proper family $\mathcal{G}=\{(W_\alpha\times I,\nu_\alpha),\lambda\}$ with respect to a map $F:M\to M$,
(a). there exists $d$-fraction of $\mathcal{G}$ that get coupled),
(b). on the complement of that fraction, there exists a measurable $\mathbb{N}$-valued function $s$, such that each component $s^{-1}(n)$ has volume $q_n$ and $F^n(s^{-1}(n))$ is again a proper family.
Then the fraction $p_n$ of coupled part after $n$ iterations satisfies
$p_n=(1-d)(q_n+(1-d)q_1\cdot p_{n-1}+\cdots+(1-d)q_{n-1}\cdot p_1)$.

Let $p(x)=\sum_{n\ge1}p_n x^n$ and $q(x)=\sum_{n\ge1}q_n x^n$. Then above relation can be restated as
$p(x)=(1-d)q(x)+(1-d)^2\sum_{i\ge1}\sum_{n>i} q_i x^i\cdot p_{n-i}x^{n-i}$ $=(1-d)q(x)+(1-d)^2\sum_{i\ge1}q_i x^i\cdot\sum_{n>i} p_{n-i}x^{n-i}$ $=(1-d)q(x)+(1-d)^2 q(x)\cdot p(x)$.

In particular, if $q_n$ decay exponentially, then the radius of convergence of $q(x)$ is larger than $1$, so is $p(x)$. Hence $p_n$ decay exponentially, too.

2. F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, R. Ures gave some interesting applications of their Ergodic Homoclinic Class $\Lambda(p)$ [here]. An invariant measure $\mu$ is said to be SRB if the following two conditions hold
(a). $\chi^1(x)>0$ for $\mu$-a.e. $x$ (which lead to $W^u(x)$ and a unstable lamination $\mathcal{W}^u$ of some full $\mu$-measure subset $E$),
(b). for any measurable partition $\xi^u\succ \mathcal{W}^u$ of $E$, the conditional measures of $\mu$ on $\xi^u(x)$ is absolutely continuous with respect to the leaf volume $m_{W^u(x)}$ for $\mu$-a.e. $x$.

Ledrappier and Young proved that $\mu$ is an SRB iff Pesin entropy formula holds: $h_\mu(f)=\int \chi^+(x)d\mu(x)$. In particular $h_\mu(f)>0$ and $\chi^s(x)<0$ for $\mu$-a.e. $x$.

Corollary 1.3 there concerns the ergodic components of SRB-measures for surface diffeomorphisms. Namely, let $f$ be a $C^2$ surface diffeomorphism and $\mu$ be an SRB-measure. If $\mu(\Lambda(p))>0$, then the conditional measure $\mu|_{\Lambda(p)}$ is ergodic.
The idea of Proof is to apply Hopf argument: the forward Birkhoff average $\phi^+(x)$ is almost constant. Consider a subset $F_{\epsilon,k}$ of the Pesin block $E_{\epsilon,k}$, consisting of leaf-density points of $E_{\epsilon,k}\cap W^u_{\epsilon}(x)$ for all $x\in E_{\epsilon,k}$. Then let $G_{\epsilon,k}$ be the set of recurrent points with respect to $F_{\epsilon,k}$. Clearly $\mu(G_{\epsilon,k})=\mu(E_{\epsilon,k})$.
(a Construct a magnet). Let $x\in G_{\epsilon,k}$. Pick a point $z\in W^s(x)\pitchfork W^u(p)$. Replacing $x$ by some $f^kx$ if necessary, we assume $z\in W^s_{\delta}(x)$ for some $\delta\ll\epsilon$. In particular $W^s_{\epsilon/2}(\hat{x})\pitchfork W^u(p)$ for $\hat{x}$ on a positive leaf-measure subset $A=W^u_{loc}(x)\cap E_{\epsilon,k}$ (Remark 1).
(b Cross the magnet). Let $y\in \Lambda(p)$ and pick a point $z'\in W^s(y)\pitchfork W^u(p)$. Then applying Inclination Lemma we see $f^nW^u_\epsilon(y)$ will converge to $W^u(p)$ and hence $f^nW^u_\epsilon(y)\pitchfork W^s_{2\delta}(x)$ for all $n$ large enough. Replacing $y$ by some $f^ky$ if necessary, we assume $W^u(y)\pitchfork W^s_{2\delta}(x)$ and hence $W^u(y)\pitchfork W^s_{\epsilon}(\hat{x})$ for every $\hat{x}\in A$. Viewing the two unstable plaques as transversals of the stable manifolds on Pesin block $E_{\epsilon,k}$, we conclude that $h(A)$ has a positive $m_{W^u(y)}$-volume since the holonomy is absolutely continuous.
(c Hopf argument). $\phi^+$ is $m_{W^u(y)}$-a.e. constant and $m_{W^u(x)}$-a.e. constant (since $\mu$ is SRB), and achieve the same constant since a positive measure subset $A$ of $W^u(x)$ is connected to and $h(A)$ of $W^u(y)$ by the Pesin stable manifolds. So $\phi^+$ has the same constant on a.e. $W^u(y)$ for all $\mu$-a.e. $y$. Then Hopf argument lead to the ergodicity of $\mu|_{\Lambda(p)}$.

Remark 1. Note that the shape of $W^u(p)$ may be quite wild near the intersection $z$. But we can take $A$ quite short to avoid the potential trouble–this is why we put the density requirement.

Observation from their proof The union $\bigcup_{A}W^s_{loc}(\hat{x})$ forms a nontrivial magnet. Each Birkhoff-regular $W^u(x)$ crossing this magnet has the same forward Birkhoff average $m_u$-a.e.

1. Bonatti and Viana proved in [here] the following theorems for partially hyperbolic attractor $(f,\Lambda)$ with $T_{\Lambda}M=E^u\oplus E^c$:
Theorem A. If $m_D\{x\in D|\chi^+_c(x) <0\} >0$ for any unstable disc $D$, then $f$ has finitely many $SRB$-measures and the union of their basins has total Lebesgue measure in the basin of attraction of $\Lambda$.
Theorem B. If $\mathcal{F}^u$ is minimal in $\Lambda$, and $m_D\{x\in D|\chi^+_c(x) <0\} >0$ for some unstable disk $D$, then there is a unique $SRB$-measure $\mu$ and $m(B(\Lambda)\backslash B(\mu,f))$.

Dolgopyat remarked [here] that Theorem B holds under a slightly weaker assumption, that almost all unstable leaves are dense, which follows (in the volume preserving case) from the accessibility property.