4. Ledrappier-Young’s criterion for measures with Absolutely Continuous Conditional measures on Unstable manifolds. Let be a diffeomorphism and be an invariant probability measure. For -a.e. , if there exist negative Lyapunov exponent(s) at , then the set of points with exponentially approximating future of is a -submanifold, . Then is said to have ACCU if for each measurable partition with and contains an unstable plaque for -a.e. , the conditional measure for -a.e. .

Theorem. has ACCU if and only if , where .

Definition. Such a measure is called the SRB measure.

Moreover, the density is strictly positive and on for -a.e. .

3. Exponential tail bound for coupling lemma. Suppose that for each proper family with respect to a map ,

(a). there exists -fraction of that get coupled),

(b). on the complement of that fraction, there exists a measurable -valued function , such that each component has volume and is again a proper family.

Then the fraction of coupled part after iterations satisfies

.

Let and . Then above relation can be restated as

.

In particular, if decay exponentially, then the radius of convergence of is larger than , so is . Hence decay exponentially, too.

2. F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, R. Ures gave some interesting applications of their Ergodic Homoclinic Class [here]. An invariant measure is said to be SRB if the following two conditions hold

(a). for -a.e. (which lead to and a unstable lamination of some full -measure subset ),

(b). for any measurable partition of , the conditional measures of on is absolutely continuous with respect to the leaf volume for -a.e. .

Ledrappier and Young proved that is an SRB iff Pesin entropy formula holds: . In particular and for -a.e. .

Corollary 1.3 there concerns the ergodic components of SRB-measures for surface diffeomorphisms. Namely, let be a surface diffeomorphism and be an SRB-measure. If , then the conditional measure is ergodic.

The idea of Proof is to apply Hopf argument: the forward Birkhoff average is almost constant. Consider a subset of the Pesin block , consisting of leaf-density points of for all . Then let be the set of recurrent points with respect to . Clearly .

(a Construct a magnet). Let . Pick a point . Replacing by some if necessary, we assume for some . In particular for on a positive leaf-measure subset (Remark 1).

(b Cross the magnet). Let and pick a point . Then applying Inclination Lemma we see will converge to and hence for all large enough. Replacing by some if necessary, we assume and hence for every . Viewing the two unstable plaques as transversals of the stable manifolds on Pesin block , we conclude that has a positive -volume since the holonomy is absolutely continuous.

(c Hopf argument). is -a.e. constant and -a.e. constant (since is SRB), and achieve the same constant since a positive measure subset of is connected to and of by the Pesin stable manifolds. So has the same constant on a.e. for all -a.e. . Then Hopf argument lead to the ergodicity of .

Remark 1. Note that the shape of may be quite wild near the intersection . But we can take quite short to avoid the potential trouble–this is why we put the density requirement.

Observation from their proof The union forms a nontrivial magnet. Each Birkhoff-regular crossing this magnet has the same forward Birkhoff average -a.e.

1. Bonatti and Viana proved in [here] the following theorems for partially hyperbolic attractor with :

Theorem A. If for any unstable disc , then has finitely many -measures and the union of their basins has total Lebesgue measure in the basin of attraction of .

Theorem B. If is minimal in , and for some unstable disk , then there is a unique -measure and .

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.