Tangent Space of the Stable And Unstable Manifold of Anosov Diffeomorphism on 2-Torus

TL;DR

This paper provides an explicit formula for tangent vectors of stable and unstable manifolds of certain Anosov diffeomorphisms on the 2-torus, using formal series and derivative trees, including perturbations of hyperbolic automorphisms.

Contribution

It introduces a method to explicitly compute tangent vectors of manifolds for perturbed Anosov diffeomorphisms on the 2-torus using formal series and derivative trees.

Findings

Explicit tangent vector formulas for perturbed systems

Extension of linear automorphism analysis to nonlinear perturbations

Method applicable to a class of Anosov diffeomorphisms on the 2-torus

Abstract

In this paper we describe the tangent vectors of the stable and unstable manifold of a class of Anosov diffeomorphisms on the torus $\mathbb{T}^2$ using the method of formal series and derivative trees. We start with linear automorphism that is hyperbolic and whose eigenvectors are orthogonal. Then we study the perturbation of such maps by trigonometric polynomial. It is known that there exist a (continuous) map $H$ which acts as a change of coordinate between the perturbed and unperturbed system, but such a map is in general, not differentiable. By "re-scaling" the parametrization $H$, we will be able to obtain the explicit formula for the tangent vectors of these maps.