Abstract

In the context of volume-dissipative surface diffeomorphisms, we present a renormalization scheme that decomposes the non-wandering set into homoclinic classes and "odometers." We further establish a weak stability property for this decomposition under small perturbations, assuming no bifurcations occur. This result offers a non-uniformly hyperbolic counterpart to Steve Smale's well-known Axiom A spectral decomposition and Omega-stability theorems.