Finite Time Hyperbolic Coordinates
Stefano Luzzatto, Dominic Veconi, Khadim War
TL;DR
This work introduces finite-time hyperbolic coordinates by formalizing co-eccentricity as a finite-time analogue of hyperbolicity and defining hyperbolic coordinates $\{e^{(k)},f^{(k)}\}$ from the major/minor axes of the ellipse $D\Phi^{k}_{\xi_{0}}(\mathcal{S}_{0})$. It then defines quasi-hyperbolicity to capture pointwise, finite-time hyperbolic behavior in both non-singular and singular settings, including maps with unbounded derivatives, and proves convergence and derivative bounds for these coordinates. The key contributions are explicit exponential convergence rates for the hyperbolic directions under Type I/II quasi-hyperbolicity, uniform bounds on the derivatives of hyperbolic coordinates with respect to the base point, and a detailed a priori/a posteriori framework that handles slow variation of the finite-time foliations. These results provide a robust geometric-analytic toolbox for nonuniform and singular hyperbolic systems, with potential applications to SRB measures and extensions of Benedicks–Carleson-type arguments to broader classes (e.g., Henon-type maps, Lorenz-like maps) where classical uniform hyperbolicity fails. Overall, the paper advances a tractable finite-time approach to hyperbolicity, enabling precise control of the most contracted directions and their variation along orbits in systems beyond strong dissipativity.
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.