Can we know if a map is hyperbolic without making any calculation?

Question

Can we know if a map is hyperbolic without making any calculation, i.e without verifying if the differential of the map has or has not eigenvalues of absolute values 1?

P.S. Can we show that the spatial version of Baker map is hyperbolic only by the construction steps?

Answer 1

Let $f : M \to M$ be a $C^1$ diffeomorphism on a Riemannian manifold $M$, and let $\Lambda \subset M$ be an invariant subset (i.e., $f (\Lambda) = \Lambda$).

We say that $f$ is hyperbolic on $\Lambda$ if, for each $x \in \Lambda$, there is a splitting $T_x M = E_x^u \oplus E^s_x$ with the following properties:

1. For all $x \in \Lambda$, we have $df_x E^{u/s}_x = E^{u/s}_{f x}$.
2. There are constants $C > 0, \lambda \in (0,1)$ such that $$ \| df^n_x |_{E^s_x} \| \leq C \lambda^n \, , \quad \text{and} \quad \| df^{-n}_x |_{E^u_x} \| \leq C \lambda^n $$ for all $x \in \Lambda$ and $n \geq 0$.

Unless the mapping in question has a very simple structure (as is the case, for instance, for the Baker's transformation) it is often somewhat challenging to find $E^{u/s}_x$ explicitly. In practice, one looks not for $E^{u/s}_x$ but rather for families of stable/unstable cones.

Theorem: $f$ is hyperbolic on $\Lambda$ iff the following holds.

For each $x \in \Lambda$ there is a splitting $T_x M = \tilde E^u_x \oplus \tilde E^s_x$ and numbers $\alpha > 0, C > 0, \lambda \in (0,1)$ with the following properties. For each $x \in \Lambda$, define \begin{gather*} \mathcal C^u_x := \{ v \in T_x M : \| \pi^s_x v\| \leq \alpha \| \pi^u_x v\|\} \\ \mathcal C^s_x := \{ v \in T_x M : \| \pi^u_x v\| \leq \alpha \| \pi^s_x v\|\} \end{gather*} For each $x \in\Lambda$,

1. $df_x \mathcal C^u_x \subset \operatorname{Interior}(\mathcal C^u_{fx})$ and $df^{-1}_x \mathcal C^s_x \subset \operatorname{Interior}(\mathcal C^s_{f^{-1}x})$

2. For each $v \in \mathcal C^u_x$, we have $\| df_x^n v\| \geq C^{-1} \lambda^{-n}\| v\|$ and for each $w \in \mathcal C^s_x$, we have $\| df^n_x w\| \leq C \lambda^n \| w\|$.

A good reference is the book of Brin and Stuck. Notice that in the cones definition, the subspaces $\tilde E^{u/s}_x$ need not be $df$-invariant. This makes the above alternative definition much more convenient for applications.

To get a feeling for this alternative definition, you might consider checking the above equivalence holds when $\Lambda = \{ x\}$ is just a fixed point.

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An aside: Notice that I didn't reference eigenvalues or eigenspaces in either definition of hyperbolicity. The reason is that eigenvalues are only defined for endomorphisms of a single vector space, not between mappings of different vector spaces, e.g., the differential $df_x : T_x M \to T_{fx} M$. There is, however, a spectral approach to defining hyperbolicity using Mather's evolution operator formulation-- see, e.g., the book of Chicone and Latushkin.