Some of the conclusions in the post are misleading. For systems that are uniformly expanding the statistical description of the dynamical system is very stable, and such systems can have sense periodic orbits that are necessarily all unstable! By statistical description I mean (at least) the long term frequency that the system is in any particular region of state space (i.e. the systems physical invariant measure).<p>It is actually one of the nice facts of life that as you dial up the local instability of a dynamical system you get a corresponding increase in the statistical stability. As systems become more and more chaotic they act more and more like iid coin tosses, which we understand quite well.