Tangent bundles to regular basic sets in hyperbolic dynamics

TL;DR

This paper constructs an invariant tangent bundle to local unstable manifolds for hyperbolic sets in dynamical systems, providing a new geometric structure that approximates the set.

Contribution

It introduces a method to build an invariant tangent bundle to unstable manifolds for hyperbolic sets with $C^1$ laminations, advancing geometric understanding.

Findings

Invariant continuous tangent bundle constructed

Provides local approximation of hyperbolic set

Enhances geometric analysis of hyperbolic dynamics

Abstract

Given a locally maximal compact invariant hyperbolic set $\Lambda$ for a $C^1$ flow or diffeomorphism on a Riemann manifold with $C^1$ unstable laminations, we construct an invariant continuous bundle of tangent vectors to local unstable manifolds that locally approximates $\Lambda$ in a certain way.