Exotic Attractors pp 35-64 | Cite as

# Liapunov Stability and Adding Machines

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## 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.

## Keywords

Periodic Orbit Periodic Point Inverse Limit Symbolic Dynamic Topological Conjugacy## Preview

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