Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds

TL;DR

This paper extends the concept of hyperbolic sets to discrete dynamical systems on pseudo-Riemannian manifolds, establishing conditions for tangent space decomposition and analyzing a global attractor in hyperbolic space.

Contribution

It introduces a new definition of hyperbolic sets in pseudo-Riemannian contexts and proves the continuity of tangent space decomposition under this framework.

Findings

Decomposition of tangent space into stable and unstable subspaces with exponential dynamics.

Continuity of the tangent space decomposition via pseudo-Riemannian connection.

Identification of a global attractor in hyperbolic space not forming a hyperbolic set.

Abstract

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part of tangent space (at each point of this set) to two unstable and stable subspaces with exponentially increasing and exponentially decreasing dynamics on them. We prove the continuity of this decomposition via the metric created by a torsion-free pseudo-Riemannian connection. We present a global attractor for a diffeomorphism on an open submanifold of the hyperbolic space $H^{2}(1)$ which is not a hyperbolic set for it.