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.


New notes

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.

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