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Liapunov stability and adding machines. (English) Zbl 0848.54027
Let \(X\) be a locally compact metric space, \(f : X \to X\) be a continuous map, a set \(A \subset X\) be compact and transitive under \(f\) (i.e., there is a point in \(A\) whose \( \omega\)-limit set is the set \(A\). Let \(x \sim y\) iff \(x,y\) belong to the same connected component of \(A\). Consider the quotient space \(K = A/ \sim\) with the identification topology and the continuous map \(\widetilde f : K \to K\) induced by \(f\).
First, the authors give a proof of the ‘folklore’ result saying that either \(K\) is finite and \(\widetilde f\) is a cyclic permutation of \(K\) or \(K\) is a Cantor set and \(\widetilde f\) is transitive on \(K\). The authors are further interested in what happens if the set \(A\) is assumed to be Lyapunov stable or asymptotically stable (i.e., Lyapunov stable and simultaneously attracting).
The main result of the paper states that if, additionally, \(X\) is locally connected and \(A\) is Lyapunov stable and if \(A\) has infinitely many components (i.e., \(K\) is a Cantor set) then \(\widetilde f : K \to K\) is topologically conjugate to a (generalized) adding machine. A number of consequences of this result are derived, including a complete classification of compact Lyapunov stable transitive sets for continuous maps of the interval and the Lyapunov instability of the invariant minimal Cantor set for any Denjoy map of the circle.
Then the authors prove that, contrary to Lyapunov stability, the asymptotic stability of \(A\) implies that \(A\) has finitely many components, i.e., \(K\) is finite and \(\widetilde f\) acts on \(K\) as a cyclic permutation.
Since \(\widetilde f\) does not uniquely determine an adding machine to which it is topologically conjugate, the question of classification of adding machines up to topological conjugacy arises. This problem is also solved in the paper. Finally, given any adding machine it is shown that a homeomorphism of the closed disk \(D \subset \mathbb{R}^2\) for which the dynamics of \(\widetilde f\) on \(K\) is topologically conjugate to that of the adding machine can be constructed.

54H20 Topological dynamics (MSC2010)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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