Liapunov Stability and Adding Machines

Abstract

In Chapter 1 we discussed several notions of stability for compact invariant sets of dynamical systems. Here we shall prove that, under very general hypotheses, the set of connected components of a stable set of a discrete dynamical system possesses a tightly constrained structure. More precisely, suppose that X is a locally compact, locally connected metric space, f: X → X is a continuous mapping (not necessarily invertible) and A is a compact transitive set. Let $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f}$ K be the set of connected components of A and let : K → K be the map induced by f We proved in § 1.3 that either K is finite or a Cantor set; in either case f acts transitively on K. Our main result (Theorem 2.3.1 below) is that, if A is Liapunov stable and has infinitely many connected components, then $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f}$ acts on K as a ‘generalized adding machine’, which we describe in a moment. We remark that imposing the stronger condition of asymptotic stability destroys the Cantor structure altogether and K must be finite — which is the content of Theorem 1.4.6. Thus adding machines can be Liapunov stable but never asymptotically stable. This Theorem may be strengthened to a version that does not require transitivity but the weaker property of being a stable ω-limit set.