The Stability of Pointwise Hyperbolic Systems

TL;DR

This paper investigates the stability of pointwise hyperbolic systems on Riemannian manifolds, introducing new properties and lemmas to establish their stability even when hyperbolicity weakens near boundaries.

Contribution

It constructs the expansive property and shadowing lemma for pointwise pseudo orbits, advancing understanding of stability in systems with variable hyperbolicity.

Findings

Established the expansive property for pointwise hyperbolic systems

Proved the shadowing lemma for pointwise pseudo orbits

Demonstrated stability of systems with weakening hyperbolicity near boundaries

Abstract

The stability of the system is an important part of the research on differential dynamical systems. This paper considers a pointwise hyperbolic system defined on a connected open subset N of a compact smooth Riemannian manifold M. The hyperbolicity may weaken when approaching the boundary of the open set. By analogy with the stability of hyperbolic systems, this paper constructs the expansive property and the shadowing lemma on the pointwise pseudo orbits and thus obtains the stability of pointwise hyperbolic systems.