Hyperbolicity versus weak periodic orbits inside homoclinic classes
Dynamical Systems
2017-09-27 v2
Abstract
We prove that, for -generic diffeomorphisms, if the periodic orbits contained in a homoclinic class have all their Lyapunov exponents bounded away from 0, then must be (uniformly) hyperbolic. This is in sprit of the works of the stability conjecture, but with a significant difference that the homoclinic class is not known isolated in advance, hence the "weak" periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an "intrinsic" nature, and the classical proof of the stability conjecture does not pass through. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.
Cite
@article{arxiv.1504.03153,
title = {Hyperbolicity versus weak periodic orbits inside homoclinic classes},
author = {Xiaodong Wang},
journal= {arXiv preprint arXiv:1504.03153},
year = {2017}
}