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 either \Lambda_U(f) is not a uniformly hyperbolic set, or it’s hyperbolic but  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.

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: