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
.
—————————————————————–
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.