Differentiable normal linearization of partially hyperbolic dynamical systems

TL;DR

This paper establishes a differentiable local conjugacy for partially hyperbolic diffeomorphisms, linearizing the hyperbolic component without non-resonance conditions and improving previous $C^0$ results.

Contribution

It introduces a semi-decoupling method and Lyapunov-Perron equations to achieve optimal $C^{1,eta}$ linearization without non-resonance assumptions.

Findings

Achieves $C^{1,eta}$ linearization of hyperbolic components

Removes non-resonance conditions required in classical theorems

Improves upon previous $C^0$ linearization results

Abstract

A result on $C^0$ linearization which is differentiable at the hyperbolic fixed point is known. In this paper, we further investigate a partially hyperbolic diffeomorphism $F$ to find a local $C^0$ conjugacy, which is $C^1$ on the center manifold, to linearize the hyperbolic component (normal to the center direction) and obtain its Takens' normal form. Our result is optimal, as it needs no non-resonant condition usually required for smooth conjugacy (e.g., as in the Takens' theorem) and the $C^{1,\alpha}$ $(\alpha>0)$ smoothness condition is sharp. For the proof, the center direction obstructs the decoupling of $F$ as the stable and unstable foliations do not intersect. We overcome this difficulty via a semi-decoupling method only with the unstable foliation, where a modified Lyapunov-Perron equation needs to be established along the center direction. Subsequent issues of cocycle reduction and differentiable linearization for an expansive fiber-preserving mapping are then addressed by the Whitney's extension theory and a lifting technique, respectively. In the local context, our result improves the result of $C^0$ normal linearization by [C. Pugh and M. Shub, Invent. Math., 10 (1970): 187-198] to a differentiable one.