Some observations of Anosov systems

Let M be a closed manifold, m be the normalized volume with respect to some Riemannian metric on M, and f:M\to M be a C^2 Axiom A diffeomorphism. By Smale’s spetrum decomposition theorem (see here), the nonwandering set decomposed as \Omega(f)=\Lambda_1\sqcup\cdots\sqcup \Lambda_k (each \Lambda_i is called a basic set). Moreover the following are equivalent (see here Page 68):

\Lambda_i is an attractor;
m(W^s(\Lambda_i))>0;
P(f,\log J^u_f,\Lambda_i)=0.

Let’s further assume that f is an Anosov diffeomorphism (see here). If m(\Omega(f))>0, then m(\Lambda_i)>0 for some i. Being an attractor and repellor simultaneously, we must have \Lambda_i=M and hence f is transitive. Then for each Holder continuous function \phi:M\to\mathbb{R} there exists a unique equilibrium state \mu_{\phi}, that is, P(f,\phi)=h(f,\mu_{\phi})+\mu(\phi). In this case \mu_{\phi} is Bernoulli and

\mu_{\phi}(B(x,\epsilon,n))\sim e^{-n\cdot P(f,\phi)+(S_f^n\phi)(x)} for all x\in M, n\in\mathbb{N}, where S_f^n\phi=\phi+\cdots+\phi\circ f^{n-1} and B(x,\epsilon,n)=[y:d(f^kx,f^ky)<\epsilon,0\le k\le n-1].

Such a measure is also called Gibbs state and each point in the basin B(\mu) is a transitive point of (M,f).

Next let’s consider a special potential. Note that the hyperbolic splitting TM=E^s\oplus E^u is Holder continuous. So the function \phi^u(x)=-\log J^u(x,f)=-\log\det(D_xf:E^u_x\to E^u_{fx}) is Holder continuous. It is well known that the pressure P(f,\phi^u)=0 and the equilibrium state \mu_u=\mu_{\phi^u} is an SRB meaure whose basin B(\mu,f) is of full volume. By previous observation, we see every point x\in B(\mu,f) has a dense orbit and hence m(\mathrm{Tran}_f)=1.

An equivalent characterization of SRB measure is:
the unique f–invariant measure whose conditional measures on unstable manifolds are absolutely continuous. That is, for each \mathcal{W}^u–chart W and a uniform transversal \tau, the conditional measure \mu_{W(x)}\ll m_{W(x)} for \widehat{\mu|_X}–a.e. x\in\tau, where \widehat{\mu|_X} is the push-forward of \mu|_X under the projection \pi:W\to \tau.

Moreover f is not volume preserving, \Longleftrightarrow \widehat{\mu|_X}\nsim m_\tau, \Longleftrightarrow \widehat{\mu|_X}\perp m_\tau.

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

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 M is diffeomorphic to an open ball.
Franks proved in 1970 that \pi_1(M) has polynomial growth and M is homeomorphic to an infranilmanifold provided that \pi_1(M) 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 C^r Anosov diffeomorphisms f,g:\mathbb{T}^2\to\mathbb{T}^2 are topologically conjugate and their periodic data coincide, then they are C^{r-\epsilon}–conjugate for arbitrarily small \epsilon.

De la Llave also constructed (in 1992) two Anosov diffeomorphisms on \mathbb{T}^4 with the same periodic data which are only Holder conjugate.

The case for Anosov diffeomorphisms on \mathbb{T}^3 is left open: if two C^r Anosov diffeomorphisms on \mathbb{T}^3 conjugate and with the same periodic data, will the conjugate be C^1?

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