Hadamard–perron Theorems and Effective Hyperbolicity

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.