Do Anosov flows exist on two dimensional compact manifolds?

Question

Question:

Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an Anosov flow.

Motivation:

I am new to the field of dynamical systems. It seems to be implicit in the literature that Anosov flows do not exist on two dimensional manifolds (see http://www.sciencedirect.com/science/article/pii/S0022039605002767) but I would like to know the proof.

My current understanding:

Assume $\phi$ is Anosov and argue for a contradiction. There is a splitting of the tangent space $T_x M = E_x^s \oplus E_x^u \oplus E_x^c$ at each $x\in M$, where $E_x^c$ is the one-dimensional subspace spanned by the direction of the flow, and the stable and unstable spaces have dimension independent of $x$. Thus, without loss of generality, suppose that $E_x^u = \{0\}$ and $\text{dim}(E_x^s)=1$. The differential of the vectors $v$ in $E_x^s$ satisfy $\|d\phi_tv\| \lesssim e^{-\gamma t} \|v\|$ for all $t\geq 0$ and some $\gamma >0$. I guess that I now need to use the fact that $M$ is compact to obtain a contradiction but I don't see how to proceed. Any help would be much appreciated.

Answer 1

If M is the 2-torus then every hyperbolic automorphism gives rise to an Anosov diffeomorphism. In fact the most known and used example is Arnold's CAT map, given by the hyperbolic matrix $A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$.

In this case you have $\mathbb{R}^2$ being the direct sum of a one-dimensional expanding subspace and a one-dimensional contracting subspace. The center-space is zero dimensional in this case.

Answer 2

As you noticed, the flow has to be weakly contracting, which implies that it has to strictly contract the volume form. Therefore, $Vol(\phi_t(M))<Vol(M)$ for all $t>0$. (Here I am using compactness of $M$.) This would mean that $\phi_t$ is not a diffeomorphism, a contradiction.

Answer 3

This is by a trivial dimension count.

As I suspect this is missed due to vague definitions, here is a short summary (which is not necessarily canonical for contemporary research, even though it is faithful to the original definition by Anosov, as far as I'm concerned):

If $M$ is a $C^1$ manifold (not necessarily compact) and $\mathfrak{g}$ is a $C^0$ Riemannian metric on $M$, then a $C^1$ flow $\phi_\bullet: \mathbb{R}\to \operatorname{Diff}^1(M)$ (not necessarily volume preserving) with generator $X$ (i.e. a nowhere-zero vector field on $M$ such that $\forall t\in\mathbb{R},\forall p\in M: X(\phi_t(p))=\frac{d}{dt} \phi_t(p)$) is $\mathfrak{g}$-Anosov if there is a continuous $\phi_\bullet$-invariant splitting $TM = S(\phi_\bullet) \oplus \langle X\rangle \oplus U(\phi_\bullet)$, where $\langle X \rangle$ is the rank $1$ subbundle of $TM\to M$ generated by $X$; $S(\phi_\bullet)$ is a nontrivial subbundle consisting of vectors contracting exponentially fast w/r/t/ $\mathfrak{g}$ under the action of $\phi_\bullet$ and $U(\phi_\bullet)$ is a nontrivial subbundle consisting of vector expanding exponentially fast w/r/t/ $\mathfrak{g}$ under the action of $\phi_\bullet$. $\langle X\rangle$ is the center bundle of $\phi_\bullet$, $S(\phi_\bullet)$ is the (strong-)stable bundle of $\phi_\bullet$ and finally $U(\phi_\bullet)$ is the (strong-)unstable bundle of $\phi_\bullet$. By definition both $S(\phi_\bullet)$ and $U(\phi_\bullet)$ have rank at least $1$, meaning that $TM$ has rank at least $3$, i.e. $\dim(M)\geq 3$. Thus the case $U(\phi_\bullet)_x=E^u_x=0$ is discarded by definition.

(When $M$ is compact, the dependency on $\mathfrak{g}$ can be dropped: if $M$ is compact and $\phi_\bullet$ is $\mathfrak{g}_0$-Anosov for some $C^0$ $\mathfrak{g}_0$, then it is $\mathfrak{g}$-Anosov for any $C^0$ $\mathfrak{g}$.)

Anosov originally called Anosov flows (on closed manifolds) (U)-flows (alternatively $(\Upsilon)$-flows). Here is the relevant part from his monograph Geodesic Flows On Closed Riemannian Manifolds With Negative Curvature (p. 6):

Just to tie in with the paper you cited, a $\mathfrak{g}$-Anosov flow $\phi_\bullet$ is codimension 1 if at least one of $S(\phi_\bullet)$, $U(\phi_\bullet)$ is rank $1$.

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Finally let me mention that there are generalizations to trivial stable or unstable bundles (partial hyperbolicity, beyond-uniform hyperbolicity, ...) that cover the case you were considering; the Anosov case is very strict and structured compared to these.