Partial linearization near a hyperbolic fixed point—Classical scattering

Question

I am currently reading the famous article "Universal Properties of Maps on an Interval" by Collet, Eckmann and Lanford related to the Feigenbaum–Coullet–Tresser universality. I am in particular interested in Theorem 6.3 page 236 in that article. See the article for the precise statement, but very roughly the theorem says:

Consider a transformation $T$ on an infinite dimensional space which has a hyperbolic fixed point with one unstable direction with eigenvalue $\delta$ and a codimension one stable manifold. Then there is a change of coordinates to a new system $(x,y)$ where:

• the stable manifold is given by $y=0$
• the unstable manifold is given by $x=0$
• the transformation $T$ takes the form $$ (x,y)\longmapsto (M(x,y), \delta \ y) $$ in this new coordinate system.

In other words, this realizes a linearization in the unstable direction only.

I would like to know about similar/related theorems, follow-ups, improvements, etc. that exist in the literature.

Using keyword searches etc. has been quite disappointing and I can definitely use the help of people with expertise in the area. For instance, I did not know the above paper contained such a theorem until a chance discussion with one of the authors.

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Edit with some context:

The method used in the CEL article is as follows. They first do some prep work in order to have a coordinate system $(x,y)$ satisfying the first two properties, i.e., such that the stable and unstable manifolds are straight. Then they construct the partial conjugation as $$ z(x,y)=\lim_{n\rightarrow \infty} \delta^{-n} y_n(x,y) $$ where $y_n$ denotes the $y$ coordinate of the $n$-th iterate of the point $(x,y)$ by a suitable cut-off modifcation of $T$.

This is very similar to the construction of wave operators in scattering theory.

The reason I am interested in this is because in recent joint work (see Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions with Ajay Chandra and Gianluca Guadagni) we proved the following:

Assume $T$ is analytic and has a hyperbolic fixed point $v_{*}$ with only one expanding direction with eigenvalue $\delta$. Then $$ \Psi(v,w)=\lim_{n\rightarrow \infty} T^n(v+\delta^{-n}w) $$ exists and is analytic (jointly in $w$ and the component of $v$ along the stable tangent space used in the analytic parametrization of the stable manifold). Here $v$ belongs to the stable manifold and $w$ is arbitrary but not too big. This function $\Psi$ is not a true linearization, not even a partial one such as the $z$ function of CEL but it shares some of that flavor. Namely, it satisfies the properties:

1. $T\circ\Psi(v,w)=\Psi(v,\delta\ w)$.
2. $\Psi$ takes its values in the unstable manifold.
3. $\Psi(v,w)=\Psi(v_{*},L_{v}(w))$ where $L_v$ is a $v$-dependent linear map onto the unstable tangent space.

The $\Psi$ function can be seen as the $w$ directional derivative of the $z$ function on the stable manifold. It is a "true linearization" on the unstable manifold only. I would like to know if similar results exist in the literature.

Answer 1

From a complex analytic standpoint, an updated discussion is embedded in Lyubich's paper "Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture":

http://arxiv.org/abs/math/9903201