Isolated sets, catenary Lyapunov functions and expansive systems

TL;DR

This paper develops models for isolated sets in dynamical systems, constructs hyperbolic Lyapunov functions with catenary properties, and applies these to expansive flows and homeomorphisms, providing new tools for hyperbolic dynamics analysis.

Contribution

It introduces a method to represent isolated sets as intersections of attractors and repellers and constructs hyperbolic Lyapunov functions with catenary curves, advancing hyperbolic dynamics theory.

Findings

Every isolated set can be obtained via a surgery as an intersection of an attractor and a repeller.

Constructed hyperbolic Lyapunov functions exhibit catenary behavior along orbits.

Established a hyperbolic metric on local cross sections for expansive flows.

Abstract

It is a paper about models for isolated sets and the construction of special hyperbolic Lyapunov functions. We prove that after a suitable surgery every isolated set is the intersection of an attractor and a repeller. We give linear models for attractors and repellers. With these tools we construct hyperbolic Lyapunov functions and metrics around an isolated set whose values along the orbits are catenary curves. Applications are given to expansive flows and homeomorphisms, obtaining, among other things, a hyperbolic metric on local cross sections for an arbitrary expansive flow on a compact metric space.