Let be a closed manifold, be the normalized volume with respect to some Riemannian metric on , and be a Axiom A diffeomorphism. By Smale’s spetrum decomposition theorem (see here), the nonwandering set decomposed as (each is called a basic set). Moreover the following are equivalent (see here Page 68):

is an attractor;

;

.

Let’s further assume that is an Anosov diffeomorphism (see here). If , then for some . Being an attractor and repellor simultaneously, we must have and hence is transitive. Then for each Holder continuous function there exists a unique equilibrium state , that is, . In this case is Bernoulli and

for all , , where and .

Such a measure is also called Gibbs state and each point in the basin is a transitive point of .

Next let’s consider a special potential. Note that the hyperbolic splitting is Holder continuous. So the function is Holder continuous. It is well known that the pressure and the equilibrium state is an SRB meaure whose basin is of full volume. By previous observation, we see every point has a dense orbit and hence .

An equivalent characterization of SRB measure is:

the unique –invariant measure whose conditional measures on unstable manifolds are absolutely continuous. That is, for each –chart and a uniform transversal , the conditional measure for –a.e. , where is the push-forward of under the projection .

Moreover is not volume preserving, , .

We have defined ergodicity for dissipative systems (see the previous post). So one might ask: for transitive Anosov , is it ergodic with respect to ?

Remark that for Anosov’s, conservativity is equivalent to ergodicity.

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There is a long-standing open conjecture attributed to John Franks that an Anosov diffeomorphism on an arbitrary compact manifold is topologically conjugate to an infra-nil automorphism. The analogy for expanding case is known:

**Theorem. **If a compact manifold M supports an expanding map, then M is homeomorphic to an infranilmanifold.

Shub showed in 1969 that the universal cover of is diffeomorphic to an open ball.

Franks proved in 1970 that has polynomial growth and is homeomorphic to an infranilmanifold provided that is virtually solvable.

Gromov showed in 1981 that any group of polynomial growth is virtually nilpotent and thus completed the proof.

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De la Llave, Marco and Moriyon proved that if two Anosov diffeomorphisms are topologically conjugate and their periodic data coincide, then they are –conjugate for arbitrarily small .

De la Llave also constructed (in 1992) two Anosov diffeomorphisms on with the same periodic data which are only Holder conjugate.

The case for Anosov diffeomorphisms on is left open: if two Anosov diffeomorphisms on conjugate and with the same periodic data, will the conjugate be ?