## Tag Archives: absolute continuous

### The volume of uniform hyperbolic sets

This is a note of some well known results. The argument here may be new, and may be complete.

Proposition 1. Let $f\in\mathrm{Diff}^2_m(M)$. Then $m(\Lambda)=0$ for every closed, invariant hyperbolic set $\Lambda\neq M$.

See Theorem 15 of Bochi–Viana’s paper. Note that Proposition 1 also applies to Anosov case, in the sense that $m(\Lambda)>0$ implies that $\Lambda=M$ and $f$ is Anosov.

Proof. Suppose $m(\Lambda)>0$ for some hyperbolic set. Then the stable and unstable foliations/laminations are absolutely continuous. Hopf argument shows that $\Lambda$ is (essentially) saturated by stable and unstable manifolds. Being a closed subset, $\Lambda$ is in fact saturated by stable and unstable manifolds, and hence open. So $\Lambda=M$.

Proposition 2. There exists a residual subset $\mathcal{R}\subset \mathrm{Diff}_m^1(M)$, such that for every $f\in\mathcal{R}$, $m(\Lambda)=0$ for every closed, invariant hyperbolic set $\Lambda\neq M$.

Proof. Let $U\subset M$ be an open subset such that $\overline{U}\neq M$, $\Lambda_U(f)=\bigcap_{\mathbb{Z}}f^n\overline{U}$, which is always a closed invariant set (maybe empty). Given $\epsilon>0$, let $\mathcal{D}(U,\epsilon)$ be the set of maps $f\in\mathrm{Diff}_m^1(M)$ that eitherĀ $\Lambda_U(f)$ is not a uniformly hyperbolic set, or it’s hyperbolic butĀ  $m(\Lambda_U(f))<\epsilon$. It follows from Proposition 1 that $\mathcal{D}(U,\epsilon)$ is dense. We only need to show the openness. Pick an $f\in \mathcal{D}(U,\epsilon)$. Since $m(\Lambda_U(f))<\epsilon$, there exists $N\ge 1$ such that $m(\bigcap_{-N}^N f^n\overline{U})<\epsilon$. So there exists $\mathcal{U}\ni f$ such that $m(\bigcap_{-N}^N g^n\overline{U})<\epsilon$. In particular, $m(\Lambda_U(g))<\epsilon$ for every $g\in \mathcal{U}$. The genericity follows by the countable intersection of the open dense subsets $\mathcal{D}(U_n,1/k)$.

The dissipative version has been obtained in Alves–Pinheiro’s paper

Proposition 3. Let $f\in\mathrm{Diff}^2(M)$. Then $m(\Lambda)=0$ for every closed, transitive hyperbolic set $\Lambda\neq M$. In particular, $m(\Lambda)>0$ implies that $\Lambda=M$ and $f$ is Anosov.

See Theorem 4.11 in R. Bowen’s book when $\Lambda$ is a basic set.

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 $\Lambda^+(\mu)=\int\sum\lambda^+_i(x)d\mu(x)$.
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. $x$.