Normally Hyperbolic Invariant Manifolds: what are the splitting rates?

Question

Let $M$ be a smooth manifold, $f: M \mapsto M$ be a $C^{r}$ diffeomorphism, for $r \geq 1$. Let $\Lambda \in M$ be a submanifold in $M$ that is invariant under the action of $F$, i.e. $F(\Lambda)=\Lambda$. Suppose that $\Lambda$ is compact. We call $\Lambda$ a normally hyperbolic invariant manifold (NHIM) if there exist a constant $C>0$, splitting rates $0< \lambda < \mu^{-1} < 1$ and a splitting of the tangent space to $M$ at each $z \in \Lambda$

$$T_{z}M = E^{s}_{z}\bigoplus E^{u}_{z} \bigoplus T_{x}\Lambda$$

such that $$v \in E^{s}_{z} \quad \Leftrightarrow \quad \mid Df^{n}(z)v \mid \leq C\lambda^{n} \mid v\mid, \quad n\geq0,$$ $$v \in E^{u}_{z} \quad \Leftrightarrow \quad \mid Df^{n}(z)v \mid \leq C\lambda^{\mid n \mid} \mid v\mid, \quad n\leq0,$$ $$v \in T_{z}\Lambda \quad \Leftrightarrow \quad \mid Df^{n}(z)v \mid \leq C\mu^{\mid n \mid} \mid v\mid, \quad n \in \mathbb{Z}.$$

My question is: how do we determine the splitting rates $\lambda$ and $\mu$ ? Do we need to ensure that $\mu$ has some apriori determined value, or can we just pick $\textbf{any}$ $\mu$ such that the above is satisfied?

Foe example, consider $F(x,y) = (2x, \frac{y}{2})$. Then $F$ has a hyperbolic fixed point at the origin with eigenvalues $\{2, \frac{1}{2} \}$. Now let us increase the phase space and add two more identity variables $(p,q)$ to obtain the map $f$:

$f(x,y,p,q,) = (2x, \frac{y}{2}, p, q)$ where $(x,y,p,q,) \in \mathbb{R}^{3} \times \mathbb{S}^{1}$. We have NHIM

$$\Lambda = \{(x,y,p,q): x = 0, \quad y = 0, \quad (p, q) \in \mathbb{R} \times \mathbb{S}^{1} \}$$

We can take $\lambda = \frac{1}{2}$ here. Taking any $v = (0,0, p, q) \in T_{z}\Lambda$, we have $| Df^{n}(z)v| = \sqrt{p^{2} + q^{2}} \leq C\mu^{\mid n \mid} \mid v\mid$.

Therefore, does it follow that we may pick $\textbf{any}$ $C \geq 1$ and $\textbf{any}$ $1 < \mu < 2$ to satisfy the above inequalities?

Answer 1

For the definition you can take any $\mu$ provided that it satisfies the inequalities. Note that the constant $C$ is not important, since one can always change the metric on $T\Lambda$ so that $C=1$ (and in fact that the $3$ bundles are pairwise orthogonal).

However, for some additional properties one may need to specify some relation between $\lambda$ and $\mu$ or, if you prefer, a spectral gap. For example, the regularity of the stable and unstable manifolds inside $\Lambda$ has to do with this relation.

How do we determine $\lambda$ and $\mu$?

Well, this depends on each particular problem and it is impossible to be so detailed (it would be like saying how one can determine a Lyapunov function for each stable solution, too much to ask for). In the general setting ultimately developed by Hirsch-Pugh-Shub it is essentially a collection of powerful results than one may try to apply in each occasion (say to foliations or laminations).

Some pointers for the origins:

For the notion that you formulate, the right names are Fenichel and independently Sacker, with a less comprehensive approach. Curiously, in their book of 1977 Hirsch-Pugh-Shub seem to suggest (not taking anything from the monumental work in the HPS book! but it is written there) that Fenichel obtained some results at the same time as they did (Fenichel's work appeared 6 years before, in 1971, as noted actually in the HPS bibliography). Note the similarity of the spectral gap condition with a related hypothesis in the Brin-Pesin work initiating the study of partial hyperbolicity in 1974, related to the regularity of intermediate foliations.