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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.

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