Invariant subsets of ACIP of partially hyperbolic diffeomorphism

4. (Notes from the paper Stable ergodicity for partially hyperbolic attractors with negative central exponents)
Let f\in\mathrm{Diff}^1(M) and L be a partially hyperbolic attractor. Then there exists a C^1 neighborhood \mathcal{U}\ni f such that every g\in\mathcal{U} possesses a partially hyperbolic attractor L_g near L. Moreover assume f_n\in\mathrm{Diff}^2(M)\to f\in\mathrm{Diff}^2(M) with Gibbs u-states \mu_n on L_n, then any weak limit is a Gibbs u-state on L.

Let \mu be an ergodic Gibbs u-state with negative central Lyapunov exponents. Then there exist an open set U such that \mu(U\Delta B(\mu))=0. 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 K over A_r\cap F^u(x,\delta) with fiber W^s(\cdot,r). Then every nearby point y\in L with Birkhoff-regular plaque F^u(y,2\delta), the intersection F^u(y,2\delta)\cap K has positive leaf volume, and some point in there must be Birkhoff-regular, say p\in W^s(q,r) for some q\in A_r\cap F^u(x,\delta). Then Hopf test: for any z\in F^u(y,2\delta), \phi_-(z)=\phi_-(p)=\phi_+(p)=\phi_+(q)=\phi_-(q)=\phi_-(x). So all Birkhoff-regular plaques lie in the same ergodic omponent.

Moreover suppose \mu is the unique Gibbs u-state of (f,L). Then there exists a C^2 neighborhood \mathcal{U}\ni f such that for every g\in\mathcal{U}, (g,L_g) possesses a unique Gibbs u-state \mu_g. Moreover \mu_g has only negative central Lyapunov exponents and \mu_g\to \mu as g\to f. So we say (f,L,\mu) is stably ergodic. Since all these measures are hyperbolic, further analysis shows that (f,L,\mu) is indeed stably Bernoulli.

The key property they listed there is: for every \delta>0, there exists r>0 and \epsilon>0 depending continuously of f such that

– for every regular point x with \chi(x)\cap[-\delta,\delta]=\emptyset, the frequency of times n such that the size of local Pesin manifolds at f^nx is larger than r is larger than \epsilon.

– Moreover, for every ergodic hyperbolic measure \mu with \chi(\mu)\cap[-\delta,\delta]=\emptyset, theand hence the set A_r of points with large Pesin manifolds has positive measure: by Kac’s formula, \displaystyle \mu(A_r)=\int\frac{1}{n}\sum_{0\le k < n}1_{A_r}(x)d\mu\ge \epsilon.

3. In the continued paper here fundamental domains have been found for many invariant subsets, in particular for the set of (Birkhoff) heteroclinic points H_f(\mu,\nu)=B(\mu,f)\cap B(\nu,f^{-1}) (see Theorem 3.2 there, where \mu\neq \nu). It is unknown if the argument can be carried out to the set of (Birkhoff) homoclinic points H_f(\mu)=B(\mu,f)\cap B(\mu,f^{-1}) (for general invariant but nonergodic measure \mu). Here is an example where there does exist a fundamental domain. Consider a flow on the plate D with spiraling source o in the center and two saddles p,q at the corners.



The second picture is from here, and is called Bowen eye-like attractor. Suppose the dynamics is symmetric and V_f(x)=\mu=\frac{\delta_p+\delta_q}{2} for every x\in D^o\backslash\{o\}, where f is the time-1 map. Then it is easy to see that there exists a fundamental domain E of B(f,\mu). 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 E turns out to be a fundamental domain of the set of (Birkhoff) homoclinic points H_{\hat f}(\mu).

2. Let f:M\to M be a C^2 partially hyperbolic diffeomorphism, \mu be an Absolutely Continuous, Invariant Probability measure. That is, the density function \phi=\frac{d\mu}{dm} is well defined in L^1(m), and the set E_\mu=\{x\in M:\phi(x)>0\} is well defined in the measure-class of \mathcal{M}(m).

It is proved (Proposition 3, here) that E_\mu is bi-essentially saturated (by a density argument). Similar argument shows that every invariant subset of E_\mu is also bi-essentially saturated. At that time I thought the classical Hopf argument can only claim the bi-essential \mu-saturation of E_\mu, 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 u-measures, which states, for example, the conditional measures \mu_{W^u(x)} of \mu with respect to the unstable foliation \mathcal{W}^u is not only abs. cont., but also smooth: the canonical density (see here) \rho^u_{\text{can}}(x,y)=\frac{d\mu_{W^u(x)}(y)}{dm_{W^u(x)}} is Holder, bounded and bounded away from zero, since ACIP is automatically a Gibbs u-measure.

So let E be an invariant subset of E_\mu. Then Hopf argument implies that

  • \mu_{W^u(x)}(E\backslash W^u(x))=0 for \mu-a.e. x\in E, or equivalently,
  • m_{W^u(x)}(E\backslash W^u(x))=0 for \mu-a.e. x\in E (by the previous observation), and moreover
  • m_{W^u(x)}(E\backslash W^u(x))=0 for m-a.e. x\in E (since \mu\simeq m on E_\mu).
  • Then a standard argument shows that E is essentially u-saturated. Similarly ACIP is automatically a Gibbs s-measure and E is essentially s-saturated. This shows that E is bi-essentially saturated by Hopf argument and Gibbs theory.

    1. Let W be a plaque of the Pesin unstable manifold of f, and consider a function \rho(x) with the property that \displaystyle \frac{\rho(x)}{\rho(y)}=\prod_{k\ge1}\frac{J^u(f,f^{-k}y)}{J^u(f,f^{-k}x)} for all x,y\in W, and the normalizing condition \int_W \rho\,dm_W=1. Let \mu=\rho m_W be the induced probability on W. It is conditionally invariant under f: Consider its pushforward f\mu=\eta m_{fW}. Then: \mu(A)=(f\mu)(fA)=\int_{fA}\eta(y) dm_{fW}(y)=\int_{A}\eta(fx)\cdot J^u(f,x)dm_W(x) for any A\subset W. Hence \rho(x)=\eta(fx)\cdot J^u(f,x). In particular \displaystyle \frac{\eta(fx)}{\eta(fy)}=\frac{\rho(x)}{\rho(y)}\cdot\frac{J^u(f,y)}{J^u(f,x)}=\frac{\rho(fx)}{\rho(fy)}.
    Then by definition, both \rho and \eta induce probabilities and must coincide:
    f(\rho\cdot m_W)=(\rho\circ f)\cdot m_{fW}. Such measures are called the leafwise u-Gibbs measures.

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