Hadamard-Perron theorems and effective hyperbolicity

TL;DR

This paper extends the Hadamard-Perron Theorem to more general settings, introduces the concept of effective hyperbolicity, and provides a finite-information closing lemma for local dynamical systems.

Contribution

It presents new versions of the Hadamard-Perron Theorem, introduces effective hyperbolicity, and proves a finite-information closing lemma for local diffeomorphisms.

Findings

Classical Hadamard-Perron Theorem is recovered as a special case.

Effective hyperbolicity ensures well-behaved local manifolds with positive frequency.

Finite orbit segments can be used to verify conditions for a closing lemma.

Abstract

We prove several new versions of the Hadamard-Perron Theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard-Perron Theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of "effective hyperbolicity" and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well-behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.