I'm currently trying to understand the Birman-Williams Template Theorem, proved in the paper "Knotted periodic orbits II: Fibered knots, Low Dimensional Topology". Unfortunately, there doesn't seem to be a free online version but there are two main ways of arguing the statement which I'll reproduce here.
Recall that an invariant set $\Lambda$ is hyperbolic with respect to flow $\phi_t$ if the tangent space restricted to $\Lambda$ can be decomposed into the unstable, stable and center bundle, i.e. $TM_{\Lambda}=E^s\oplus E^u \oplus E^c$. We assume that $M$ is a differentiable manifold of dimension 3 here. By Smale's Decomposition Theorem, we can decompose a special kind of hyperbolic invariant set $\Lambda$ known as the chain-recurrent set into finite disjoint closed subsets that were themselves invariant under $\phi_t$, i.e. $\Lambda=B_1\cup B_2 \cup ... \cup B_n$. We call the subsets $B_i$ basic sets.
In the Birman-Williams paper, they argue:
Suppose $B$ is a basic set for $\phi_t$ of dimension 2. It follows that $B$ is an attractor or repeller, as one has $u+s=2$ where $u$ (resp. $s$) is the unstable (resp. stable) dimension.
This is all they say. I don't follow this reasoning - it seems quite mysterious to me.
There's another explanation of this argument in Ghrist et. al.'s monograph on Knots and Links in 3-Dimensional Flows, where they argue:
Since $B$ is two dimensional and hyperbolic and $M$ three-dimensional, the stable, unstable, and center bundles must each be of dimension one. Since $B$ must contain the center bundle, it must also contain either the stable or unstable bundle, leaving only the remaining direction. Hence, $B$ is either an attractor or a repeller.
I don't understand why the stable, unstable and center bundles must each be of dimension one - I accept that the center bundle bundle must be of dimension one, but I don't quite get the rest. It's also not clear to me that $TM_B$ has to be of dimension 2. Intuitively, we know that the tangent bundle of an $n$-dimensional manifold has dimension 2n - while it's true that $B$ need not be a manifold, why shouldn't the same thing be true here as well?
I think the kernel of my confusion resides in the fact that I am unsure about how the notion of "dimension" plays out here. I sometimes wonder if they mean something straightforward like topological dimension, or if they mean something like fractal dimension, but that just seems to confuse things even more...