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 $C^1$?