# Partial linearization near a hyperbolic fixed point—Classical scattering

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.

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.

• This looks strikingly familiar to the Poincare Linearization theorem and Normal Forms in the theory of ODEs. Mar 20, 2013 at 0:14
• @Peter: I would not say "strikingly". My question is quite unsurprisingly related to this kind of questions simply because it explores extensions of normal form theory in the direction of weaker statements than e.g. full linearizations. Mar 20, 2013 at 0:52

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

• @Adam: I was looking at this paper yesterday but it is quite long and I have yet to see where the kind of theorem I am interested in is embedded. Mar 20, 2013 at 13:38
• I'd look at Section 6. For example, Theorem 6.3 gives a hyperbolic splitting of the tangent space. Subsequent results discuss stable and unstable manifolds. If you want an actual reduction to normal form, this does not seem to be considered, and on reflection I am not aware that such a result has been formally claimed in a complex analytic setting. Mar 20, 2013 at 14:30
• @Adam: Thanks I will look at Sec 6. Mar 20, 2013 at 16:27
• Even in a finite diimensional setting there would be the issue of possible resonances among the eigenvalues, and little is known about those numbers Mar 20, 2013 at 18:57