What's so hyperbolic about hyperbolic sets?

Question

In dynamics, we have the notion of a "hyperbolic set" for a diffeomorphism $f:M\to M$ of a Riemannian manifold. I am trying to connect this to my existing ideas surrounding the term "hyperbolic". To my understanding, $M$ itself need not be a hyperbolic manifold$^*$, but maybe there is some more distant connection$^{**}$ at work?

$^*$Consider the action of $\begin{pmatrix}2&1\\1&1\end{pmatrix}$ on the torus.

$^{**}$Heh

[EDIT] Someone wanted to know what a hyperbolic set was so I'm adding the definition here. Let $M$ be a Riemannian manifold and let $f:M\to M$ be a diffeomorphism. Then $\Lambda \subset M$ is hyperbolic if there are constants $C> 0$ and $\lambda \in (0,1)$ such that for every $x\in \Lambda$, we can write $T_xM = E^u(x) \oplus E^s(x)$. We require that $\|df^n_x v\| \le C\lambda^n \|v\|$ for $v\in E^s(x)$ and $n\ge 0$, $\|df^{-n}_x v\| \le C\lambda^n \|v\|$ for $v\in E^u(x)$ and $n\ge 0$, $df_x(E^u(x)) = E^u(f(x))$, and $df_x(E^s(x)) = E^s(f(x))$.

In English, we have expanding and contracting directions. Standard example is, give me some $A\in GL_n(\mathbb{Z})$ with no eigenvalues on the unit circle. It'll induce an automorphism of the torus $\mathbb{R}^n/\mathbb{Z}^n$. Then $\mathbb{T}^n$ is a hyperbolic set with respect to this automorphism, with the expanding directions being the sum of eigenspaces with eigenvalue $> 1$, and the contracting directions being the sum of eigenspaces with eigenvalue $< 1$.

Answer 1

Yes, the connection to hyperbolic manifolds is "more distant." First, you have to read about Anosov diffeomorphisms which, in your terminology, are defined by the condition that the hyperbolic subset is the entire manifold. Next, there is one more related notion, that of an Anosov flow where in addition to the expanding/contracting subspaces of $T_xM$ one also has a 1-dimensional neutral subspace of the tangent space, tangent to the flow itself. For instance, the suspension flow of an Anosov map is an Anosov flow.

Lastly, the classical example of an Anosov flow is the geodesic flow of a manifold is strictly negative curvature, e.g. a hyperbolic manifold.

Answer 2

Gromov's "Hyperbolic Dynamics, Markov Partitions And Symbolic Categories" (https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SymbolicDynamicalCategories.pdf) is very relevant to this matter, where he starts with the question

IS THERE A UNIFIED THEORY OF HYPERBOLICITY?

Apart from geodesic flows of negatively curved manifolds, one can consider the work of Thurston as well:

Definition: Let $g\in\mathbb{Z}_{\geq2}$. Then a homeomorphism $f$ of the closed surface $\Sigma_g$ of genus $g$ is pseudo-Anosov iff its mapping torus $\mathbb{R}\otimes_f \Sigma_g=(\mathbb{R}\times \Sigma_g)/\sim_f$, $(r,f^n(x))\sim_f (r+n,x)$ admits a hyperbolic metric.

("definition" in jest; see https://mathoverflow.net/q/15731/66883)

A good starting reference for pseudo-Anosov theory is Margalit & Farb's A Primer on Mapping Class Groups (see pp.400-401 for the above definition).

More generally one can consider the well-known trichotomy of hyperbolicity-parabolicity-ellipticity, e.g. see Rastegar's "EPH-classifications in Geometry, Algebra, Analysis and Arithmetic" (https://arxiv.org/abs/1503.07859).