Finite Time Hyperbolic Coordinates

TL;DR

This paper introduces finite-time hyperbolic coordinates that describe hyperbolicity through map distortion rather than tangent space splitting, with results on their convergence and variation, aiming to extend these techniques to complex systems.

Contribution

It defines finite-time hyperbolic coordinates based on co-eccentricity, offering a new perspective on hyperbolicity beyond classical tangent space splitting.

Findings

Proves convergence properties of finite-time hyperbolic coordinates.

Describes the geometric structure and variation of these coordinates.

Lays groundwork for extending hyperbolic analysis to complex systems.

Abstract

We define finite-time hyperbolic coordinates, describe their geometry, and prove various results on both their convergence as the time scale increases, and on their variation in the state space. Hyperbolic coordinates reframe the classical paradigm of hyperbolicity: rather than define a hyperbolic dynamical system in terms of a splitting of the tangent space into stable and unstable subspaces, we define hyperbolicity in terms of the co-eccentricity of the map. The co-eccentricity describes the distortion of unit circles in the tangent space under the differential of the map. Finite-time hyperbolic coordinates have been used to demonstrate the existence of SRB measures for the Henon map; our eventual goal is to both elucidate these techniques and to extend them to a broad class of nonuniformly and singular hyperbolic systems.