Partially hyperbolic symplectomorphism with C^1 bundles

TL;DR

The paper studies dynamical coherence for $C^2$ partially hyperbolic symplectomorphisms on four-dimensional manifolds with $C^1$ stable and unstable bundles. By exploiting the symplectic form $\omega$ and constructing the $C^1$ 1-form $\eta=i_{X^s}\omega$, it proves $\eta\wedge d\eta=0$ and uses Frobenius integrability to obtain invariant foliations tangent to $\mathbb{E}^s\oplus\mathbb{E}^c$ and $\mathbb{E}^c\oplus\mathbb{E}^u$, yielding a $C^1$ center foliation. Consequently, $f$ is dynamically coherent and the center foliation is $C^1$, with a corollary that coherence persists in a $C^1$-neighborhood of $f$. This work clarifies how the symplectic structure governs the regularity of center dynamics in four dimensions and supports broader ergodicity and classification efforts for partially hyperbolic systems.

Abstract

We prove dynamical coherence for partial hyperbolic symplectomorphism in dimension 4 whose stable and unstable bundles are C^1.

Theorems & Definitions (9)

• Theorem 1
• Corollary 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:main']}