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Differentiable normal linearization of partially hyperbolic dynamical systems

Weijie Lu, Yonghui Xia, Weinian Zhang, Wenmeng Zhang

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 $C^{1,α}$ $(α>0)$ 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.

Differentiable normal linearization of partially hyperbolic dynamical systems

Abstract

A result on linearization which is differentiable at the hyperbolic fixed point is known. In this paper, we further investigate a partially hyperbolic diffeomorphism to find a local conjugacy, which is 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 smoothness condition is sharp. For the proof, the center direction obstructs the decoupling of 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 normal linearization by [C. Pugh and M. Shub, Invent. Math., 10 (1970): 187-198] to a differentiable one.
Paper Structure (8 sections, 13 theorems, 201 equations)

This paper contains 8 sections, 13 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Main result and preliminaries
  3. Main result
  4. Some lemmas
  5. Regularity of unstable foliation
  6. Proof of main result
  7. Reduction of cocycles
  8. Linearization of expansive fiber-preserving mapping

Key Result

Lemma 1

(Ga-CMP) The mapping $F$ given in NL-F such that Df-hold-1 and sp-A hold has a $C^{1,\alpha}$ center manifold where $\varpi_c:X_c\to X_{su}$ is a $C^{1,\alpha}$ mapping such that $\varpi_c(0)=0$ and $D\varpi_c(0)=0$.

Theorems & Definitions (24)

  • Lemma 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Proposition 8
  • proof : Proof of Proposition \ref{['Fol-thm']}.
  • Lemma 9
  • ...and 14 more