Tag Archives: Anosov

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.

Generic Anosov does not admit fat horseshoe

To start let’s describe an interesting proposition in QIU Hao’s paper (Commun. Math. Phys. 302 (2011), 345–357.)

Assume $f\in\mathrm{Diff}^1(M)$ and $\Lambda$ be a basic (isolated and transitive, or mixing) hyperbolic set of $f$. It is well known (Anosov) that there exists an open neighborhood $\mathcal{U}\ni f$ and an open set $U\supset\Lambda$ such that for each $g\in\mathcal{U}$,

1). $\Lambda_g=\bigcup_{n\in\mathbb{Z}}g^nU$ is an isolated hyperbolic set of $g$. Moreover $\Lambda_g\to\Lambda$ as $g\to f$.
2). there exists a (Holder) homeomorphism $h_g:\Lambda\to\Lambda_g$ such that $h_g\circ f(x)=g\circ h_g(x)$ for every $x\in\Lambda$. Moreover $h_g\to \mathrm{Id}$ as $g\to f$.

Now let’s consider the unstable log-Jacobian $\phi_g\in C(\Lambda_g,\mathbb{R})$ as
$\phi_g(x)=-\log\det(D_xg:E^u_g(x)\to E^u_g(gx)), x\in\Lambda_g$.

By classical hyperbolic theory (Sinai, Ruelle and Bowen), we know that for each $g\in\mathcal{U}\cap \mathrm{Diff}^2(M)$, the topological pressure $P(\phi_g;g,\Lambda_g)=0$ and there exists a unique equilibrium state of $\phi_g$ with respect to $(g,\Lambda_g)$.

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Define a map $\Phi:\mathcal{U}\to C(\Lambda,\mathbb{R})$ by $\Phi(g)(x)=\phi_g(h_g(x))$. It takes a few seconds to see that $\Phi$ is continuous.

Proposition 3.1 (Qiu) For each $g\in\mathcal{U}$, $P(\phi_g;g,\Lambda_g)=0$.
Proof. Since topological pressure is invariant under topological conjugation, we have
$P(\phi_g;g,\Lambda_g) = P(\Phi(g);\Lambda,f)$.
Now we pick $g_k\in\mathcal{U}\cap \mathrm{Diff}^2(M)$ with $g_k\to f$. Note this also implies $h_{g_k}\to\mathrm{Id}$, $\Phi(g_k)\to\phi_f$. So
$P(\phi_f;f,\Lambda)=\lim_{k\to\infty}P(\Phi(g_k);f,\Lambda)=\lim_{k\to\infty}P(\phi_{g_k};g_k,\Lambda_{g_k})=0$.
This finishes the proof.

Remark: In particular for all Anosov diffeomorphisms and all Axiom A diffeomorphisms with no cycle condition, we have $P(\phi_f;f)=0$.

Proposition 3.1 (continued). for generic $g\in\mathcal{U}$, there exists a unique equilibrium state for $\phi_g$ with respect to $(g,\Lambda_g)$.
Proof. Since $(f,\Lambda)$ is expansive, the entropy map $\mathcal{M}(f,\Lambda)\to\mathbb{R},\mu\mapsto h(f,\mu)$ is upper semicontinuous and there is a residual subset $\mathcal{R}\subset C(\Lambda,\mathbb{R})$ such that each $\phi\in\mathcal{R}$ has a unique equilibrium state with respect to $(f,\Lambda)$. Since $\Phi$ is continuous, the pre-image $\Phi^{-1}(\mathcal{R})$ is

1. a $G_\delta$ set in $\mathcal{U}$ since the pre-images of open sets are open;

2. a dense set in $\mathcal{U}$ since $\Phi(g)\in\mathcal{R}$ for all $g\in\mathcal{U}\cap\mathrm{Diff}^2(M)$.

In particular $\Phi^{-1}(\mathcal{R})$ is residual in $\mathcal{U}$. The proof is complete.

We focus on a special case of QIU’s main result. Let $\mathcal{A}^r$ be the set of $C^r$ Anosov diffeomorphisms on $M$ (might be empty). For an invariant measure $\mu$, we let $B(\mu)$ be the set of points with $\frac{1}{n}\sum_{0\le k.

Theorem A (Qiu). Generic $f\in\mathcal{A}^1$ has a unique SRB measure $\mu_f$: $m(M\backslash B(\mu_f))=0$.
Indeed, $\mu_f$ is the unique equilibrium state of $\phi_f$ (hence ergodic).

Robinson and Young constructed an Anosov diffeomorphism with nonabsolutely continuous foliations, by embedding an Bowen horseshoe $\Lambda_B$ to some $f\in\mathcal{A}(\mathbb{T}^2)$. Although $f$ is transitive, every point in $\Lambda_B$ can not be a transitive point. Theorem A implies that this phenomenon fails generically:

Observation: generic $f\in\mathcal{A}^1(M)$ does not admit Bowen’s fat horseshoe.
Proof. A priori, we donot know if every Anosov is transitive. So we divide $\mathcal{A}^1(M)=\mathcal{A}_t(M)\sqcup \mathcal{A}_e(M)$ into transitive ones and exotic ones. But we know they are always structurally stable. Therefore both parts are open.

By Theorem A, we know that, for generic $f\in\mathcal{A}_t(M)$, $\mu_f$ is ergodic and fully supported. Hence every point in $B(\mu_f)$ is a transitive point. In particular every closed invariant set of $f$ has trivial volume: 0 or 1.

For maps in the exotic ones $f\in\mathcal{A}_e(M)$, at least they can be viewed as Axiom A system, and Smale’s Spectra Decomposition Theorem applies: $\Omega(f)=\Lambda_1\sqcup\cdots\sqcup\Lambda_n$, where $n=n(f)$ is locally constant. Some of them are attractors, say $A_1,\cdots, A_k$, some are repellers, say $R_1,\cdots, R_l$. Let $\mathcal{R}_0$ be the residual subset given by QIU for all repellers. Clearly $\mathcal{R}=\mathcal{R}_0\cap\mathcal{R}^{-1}_0$ is also generic. For each $f\in \mathcal{R}$, there is an SRB $\mu_{u,i}$ relative to $R_i(f)$ with $m(\bigcup_i B(\mu_{u,i}))=1$ and an SRB $\mu_{s,j}$ relative to $A_{j}$ for $f^{-1}$ with $m((\bigcup_j B(\mu_{s,j}))=1$. Incorrect conclusion. The following are void.

To derive a contradiction, suppose that there was a fat horseshoe $H$ of $f$, then

1. either $H\cap B(\mu_{u,i})=\emptyset$, (contradicts $m(\bigcup_i B(\mu_{u,i}))=1$);

2. or $H\cap B(\mu_{s,j})=\emptyset$, (contradicts $m(\bigcup_j B(\mu_{s,j}))=1$);

3. or there exist $x\in H\cap B(\mu_{u,i})$ and $y\in H\cap B(\mu_{s,j})$. In particular $H$ has nontrivial intersections with the attarctor $A_j$ and the repeller $R_i$ simultaneously, which contradict the transitivity of Horseshoe. Q.E.D.

Katok’s entropy conjecture

In dynamics there are several conjectures related to entropy. The first is the famous Shub Entropy Conjecture:

Let $f:M\to M$ be a differentiable map (or a diffeomorphism), $s(f_*)$ be the spectral radius of the linear map $f_{*}:\oplus H_k(M,\mathbb{R})\to \oplus H_k(M,\mathbb{R})$. Then
$h_{\mathrm{top}}(f)\ge s(f_*)$.
Moveover find characteristics of maps satisfying $h_{\mathrm{top}}(f)= s(f_*)$ in each homotopy class.

I am not sure whether the following special problem is still open (it is proved if $E^u$ is orientable):

If $f:M\to M$ is an Anosov diffeomorphism, then $h_{\mathrm{top}}(f)= s(f_*)$.

There is another entropy conjecture due to Katok. In fact Katok proved that the topological entropy and the Liouville entropy of a geodesic flow on a negatively curved surface agree only if the curvature is constant, that is, the metric is locally symmetric. He conjectured that the same holds in any dimension. One of the best result in this direction is

Besson, Courtois and Gallot, the topological entropy of geodesic flow is minimized only for locally symmetric metrics.