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

Assume and be a basic (isolated and transitive, or mixing) hyperbolic set of . It is well known (Anosov) that there exists an open neighborhood and an open set such that for each ,

1). is an isolated hyperbolic set of . Moreover as .

2). there exists a (Holder) homeomorphism such that for every . Moreover as .

Now let’s consider the *unstable log-Jacobian* as

.

By classical hyperbolic theory (Sinai, Ruelle and Bowen), we know that for each , the topological pressure and there exists a unique equilibrium state of with respect to .

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Define a map by . It takes a few seconds to see that is continuous.

**Proposition 3.1** (Qiu) For each , .

Proof. Since topological pressure is invariant under topological conjugation, we have

.

Now we pick with . Note this also implies , . So

.

This finishes the proof.

**Remark**: In particular for all Anosov diffeomorphisms and all Axiom A diffeomorphisms with no cycle condition, we have .

**Proposition 3.1 **(continued). for generic , there exists a unique equilibrium state for with respect to .

Proof. Since is expansive, the entropy map is upper semicontinuous and there is a residual subset such that each has a unique equilibrium state with respect to . Since is continuous, the pre-image is

1. a set in since the pre-images of open sets are open;

2. a dense set in since for all .

In particular is residual in . The proof is complete.

We focus on a special case of QIU’s main result. Let be the set of Anosov diffeomorphisms on (might be empty). For an invariant measure , we let be the set of points with .

**Theorem A** (Qiu). Generic has a unique SRB measure : .

Indeed, is the unique equilibrium state of (hence ergodic).

Robinson and Young constructed an Anosov diffeomorphism with nonabsolutely continuous foliations, by embedding an Bowen horseshoe to some . Although is transitive, every point in can not be a transitive point. Theorem A implies that this phenomenon fails generically:

**Observation:** generic does not admit Bowen’s fat horseshoe.

Proof. A priori, we donot know if every Anosov is transitive. So we divide 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 , is ergodic and fully supported. Hence every point in is a transitive point. In particular every closed invariant set of has trivial volume: 0 or 1.

For maps in the exotic ones , at least they can be viewed as Axiom A system, and Smale’s Spectra Decomposition Theorem applies: , where is locally constant. Some of them are attractors, say , some are repellers, say . Let be the residual subset given by QIU for all repellers. Clearly is also generic. For each , there is an SRB relative to with and an SRB relative to for with . Incorrect conclusion. The following are void.

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

1. either , (contradicts );

2. or , (contradicts );

3. or there exist and . In particular has nontrivial intersections with the attarctor and the repeller simultaneously, which contradict the transitivity of Horseshoe. Q.E.D.