4. (Notes from the paper Stable ergodicity for partially hyperbolic attractors with negative central exponents)

Let and be a partially hyperbolic attractor. Then there exists a neighborhood such that every possesses a partially hyperbolic attractor near . Moreover assume with Gibbs u-states on , then any weak limit is a Gibbs u-state on .

Let be an ergodic Gibbs u-state with negative central Lyapunov exponents. Then there exist an open set such that . The analog doesn’t hold for Gibbs u-states with positive central Lyapunov exponents, since the stable and unstable directions play different roles in dissipative systems.

Proof. We build a magnet over with fiber . Then every nearby point with Birkhoff-regular plaque , the intersection has positive leaf volume, and some point in there must be Birkhoff-regular, say for some . Then Hopf test: for any , So all Birkhoff-regular plaques lie in the same ergodic omponent.

Moreover suppose is the unique Gibbs u-state of . Then there exists a neighborhood such that for every , possesses a unique Gibbs u-state . Moreover has only negative central Lyapunov exponents and as . So we say is stably ergodic. Since all these measures are hyperbolic, further analysis shows that is indeed stably Bernoulli.

The key property they listed there is: for every , there exists and depending continuously of such that

– for every regular point with , the frequency of times such that the size of local Pesin manifolds at is larger than is larger than .

– Moreover, for every ergodic hyperbolic measure with , theand hence the set of points with large Pesin manifolds has positive measure: by Kac’s formula, .

3. In the continued paper here fundamental domains have been found for many invariant subsets, in particular for the set of (Birkhoff) heteroclinic points (see Theorem 3.2 there, where ). It is unknown if the argument can be carried out to the set of (Birkhoff) homoclinic points (for general invariant but nonergodic measure ). Here is an example where there does exist a fundamental domain. Consider a flow on the plate with spiraling source in the center and two saddles at the corners.

The second picture is from here, and is called Bowen eye-like attractor. Suppose the dynamics is symmetric and for every , where is the time-1 map. Then it is easy to see that there exists a fundamental domain of . We can blow up the center, identify the corresponding boundaries of two copies and reverse the flow direction on the second copy. Then the subset turns out to be a fundamental domain of the set of (Birkhoff) homoclinic points .

2. Let be a partially hyperbolic diffeomorphism, be an Absolutely Continuous, Invariant Probability measure. That is, the density function is well defined in , and the set is well defined in the measure-class of .

It is proved (Proposition 3, here) that is bi-essentially saturated (by a density argument). Similar argument shows that every invariant subset of is also bi-essentially saturated. At that time I thought the classical Hopf argument can only claim the bi-essential -saturation of , and Proposition 3 might be out of the range of Hopf argument. Now it seems this is not the case if we combine some results in Gibbs -measures, which states, for example, the conditional measures of with respect to the unstable foliation is not only abs. cont., but also smooth: the canonical density (see here) is Holder, bounded and bounded away from zero, since ACIP is automatically a Gibbs -measure.

So let be an invariant subset of . Then Hopf argument implies that

for -a.e. , or equivalently,
for -a.e. (by the previous observation), and moreover
for -a.e. (since on ).
Then a standard argument shows that is essentially -saturated. Similarly ACIP is automatically a Gibbs -measure and is essentially -saturated. This shows that is bi-essentially saturated by Hopf argument and Gibbs theory.

1. Let be a plaque of the Pesin unstable manifold of , and consider a function with the property that for all , and the normalizing condition . Let be the induced probability on . It is conditionally invariant under : Consider its pushforward . Then: for any . Hence . In particular .

Then by definition, both and induce probabilities and must coincide:

. Such measures are called the leafwise u-Gibbs measures.