# Proper families for Anosov flows

So I've been skimming Bowen's 1972 paper "Symbolic Dynamics for Hyperbolic Flows" hoping it would give me some insight into how to build a Markov family for the cat flow (i.e., the Anosov flow obtained by suspension of the cat map with unit height). For the sake of completeness, the cat flow $\phi$ is obtained as follows:

i. Consider the cat map $A$ on the 2-torus and identify points $(Ax,z)$ and $(x,z+1)$ to obtain a 3-manifold $M$

ii. Equip $M$ with a suitable metric (e.g., $ds^2 = \lambda_+^{2z}dx_+^2 + \lambda_-^{2z}dx_-^2 + dz^2$, where $x_\pm$ are the expanding and contracting directions of $A$ and $\lambda_\pm$ are the corresponding eigenvalues.)

iii. Consider the flow generated by the vector field $(0,1)$ on $M$--that's the cat flow.

Unfortunately I'm getting stuck at the first part of Bowen's quasi-constructive proof, which requires finding a suitable set of disks and subsets transverse to the flow. Rather than rehash the particular criteria for a set of disks and subsets used in Bowen's construction, I will relay a simpler but very similar set of criteria, for a proper family (which if it meets some auxiliary criteria is also a Markov family):

$\mathcal{T} =$ {$T_1,\dots,T_n$} is called a proper family (of size $\alpha$) iff there are differentiable closed disks $D_j$ transverse to the flow s.t.

1. the $T_j$ are closed
2. $M = \phi_{[-\alpha, 0]}\Gamma(\mathcal{T})$, where $\Gamma(\mathcal{T}) = \cup_j T_j$
3. $\dim D_j = \dim M - 1$
4. diam $D_j < \alpha$
5. $T_j \subset$ int $D_j$ and $T_j = \bar{T_j^*}$ where $T_j^*$ is the relative interior of $T_j$ in $D_j$
6. for $j \ne k$, at least one of the sets $D_j \cap \phi_{[0,\alpha]}D_k$, $D_k \cap \phi_{[0,\alpha]}D_j$ is empty.

I've been stuck on even constructing such disks and subsets (let alone where the subsets are rectangles in the sense of hyperbolic dynamics). Bowen said this sort of thing is easy and proceeded under the assumption that the disks and subsets were already in hand. I haven't found it to be so. The thing that's killing me is 6, otherwise neighborhoods of the Adler-Weiss Markov partition for the cat map would fit the bill and the auxiliary requirements for the proper family to be a Markov family.

I've really been stuck in the mud on this one, could use a push.

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Just a wild guess, maybe this can help you arxiv.org/PS_cache/arxiv/pdf/0810/0810.5269v1.pdf ? they construct here a lot of Markov partitions for hyperbolic maps of 2-dimensional torus. –  Dmitri Nov 30 '09 at 22:59
Thanks, but no dice. Markov partitions of 2D hyperbolic toral automorphisms aren't hard for me to come by. I'm frankly surprised that I've been unable to adapt such a partition to a Markov family for the corresponding flow. –  Steve Huntsman Dec 1 '09 at 1:22

Take Adler-Weiss on $0\times\mathbb T^2$, $1/3\times\mathbb T^2$ and $2/3\times\mathbb T^2$. Take neighborhoods of this tripled Adler-Weiss. Then this collection would satisfy all the properties with $\alpha=1/3$.

I am not sure why are you particularly interested in suspension flow, everything is determined by the base Anosov diffeo.

Edit: Indeed, this has to be tinkered a bit. Say Adler Weiss has two rectangles with neighborhoods $D_1$ and $D_2$. Then take collection $0\times D_1$, $1/3\times D_1$, $2/3\times D_1$, $\varepsilon\times D_2$, $1/3+\varepsilon\times D_2$, $2/3+\varepsilon\times D_2$.

To ensure the second property take $\alpha=1/3+\varepsilon$.

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Unless I'm mistaken, this violates 6) because the neighborhoods at a given z-value have nonempty intersections. –  Steve Huntsman Dec 1 '09 at 1:13
Also it appears to me that staggering the rectangles along the z-direction in the putative solution above doesn't work, because then the flow doesn't cover M. –  Steve Huntsman Dec 1 '09 at 1:18
See comment above from 6 minutes before your edit. –  Steve Huntsman Dec 1 '09 at 1:55
Re: $\alpha = 1/3 + \epsilon$: consider the set $[0,\epsilon] \times D_2$. This set will have a point not in the image of $(2/3 + \epsilon) \times D_2$ under $\phi_{[0,\alpha]}$ –  Steve Huntsman Dec 1 '09 at 2:45
But it will be in the image of $2/3\times D_1$. –  Andrey Gogolev Dec 1 '09 at 3:00

Don't upvote this. I just figured since this is getting some attention from another question on MO I'd communicate the blurb that is going to go into what I'm writing now. (See also a related bit on meta.) A comment won't allow TeX at this point, hence the use of an answer.

Given a Markov partition $\mathcal{R} > =$ {$R_1,\dots, R_n$} for the cat map and $m \ge 3$, consider the sets $R_{jk} := R_j \times$ {$\frac{k}{m} - > j\epsilon$}, where $1 \le k \le m$ and $\epsilon < \frac{1}{mn}$. The family $\mathcal{R}' :=$ {$R_{jk}$} is readily seen to be a proper family for the cat flow. [A. Gogolev, private communication] The Poincaré map for $\mathcal{R}'$ sends $R_{jk}$ to $R_{j,k+1}$ for $1 > \le k \le m-1$, and because $\mathcal{R}$ is a Markov partition for the cat map it follows that $\mathcal{R}'$ is a Markov family for the cat flow.

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