Some estimates for the stable manifold theorem

TL;DR

This paper provides detailed estimates on the size and smoothness of local stable manifolds in the context of partially hyperbolic singularities, with explicit constants and applications to vector fields vanishing on submanifolds.

Contribution

It offers new quantitative estimates for the stable manifold theorem, including bounds on neighborhood size and derivatives, with explicit dependence on the vector field.

Findings

Estimates on the size of neighborhoods where stable manifolds are graphs

Bounds on derivatives of all orders of the stable manifold functions

Application to vector fields vanishing on submanifolds and controlling stable foliation charts

Abstract

We investigate the standard stable manifold theorem in the context of a partially hyperbolic singu-larity of a vector field depending on a parameter. We prove some estimates on the size of the neighbourhood where the local stable manifold is known to be the graph of a function, and some estimates about the derivatives of all orders of this function. We explicitate the different constants arising and their dependance on the vector field. As an application, we consider the situation where a vector field vanishes on a submanifold N and contracts a direction transverse to N. We prove some estimates on the size of the neighbourhood of N where there are some charts straightening the stable foliation while giving some controls on the derivatives of all orders of the charts.