The Stability of Pointwise Hyperbolic Systems
Haiye Guo, Yunhua Zhou
TL;DR
This work addresses stability for pointwise hyperbolic systems defined on a connected open set $N$ within a compact manifold $M$, where hyperbolicity weakens near the boundary. It extends classical stability concepts by formulating $\varepsilon$-pointwise pseudo orbits, a pointwise shadowing lemma, and pointwise expansivity, and uses the graph transform to construct local stable/unstable manifolds on pointwise pseudo orbits. The authors demonstrate that pointwise hyperbolicity is preserved under small interior perturbations and construct a continuous map $h$ with $h\circ g=f\circ h$, yielding semi-pointwise and, under additional assumptions, pointwise quasi-stability (with $h$ possibly a homeomorphism). These results generalize structural stability notions to nonuniform, boundary-influenced hyperbolic dynamics and provide a framework for stability analysis in pointwise hyperbolic systems.
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.