Hadamard-Perron theorems and effective hyperbolicity

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.

45 pages

Cited by in corpus (4)

• The geometric approach for constructing Sinai-Ruelle-Bowen measures
• Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps
• On mixing diffeomorphisms of the disk
• An effective version of Katok's horseshoe theorem for conservative $C^2$ surface diffeomorphisms