isolated hyperbolic set | Pengfei's Notes

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 $\frac{1}{n}\sum_{0\le k<n}\delta_{f^kx}\to \mu$ .

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.

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Generic Anosov does not admit fat horseshoe

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