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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.

##### MSC:
 54H20 Topological dynamics (MSC2010) 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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##### References:
 [1] Newhouse, Dynamical Systems, CIME lectures, Bressanone (1980) [2] DOI: 10.2307/2373810 · Zbl 0355.58010 · doi:10.2307/2373810 [3] DOI: 10.1088/0951-7715/4/3/010 · Zbl 0737.58043 · doi:10.1088/0951-7715/4/3/010 [4] Fuchs, Infinite Abelian Groups 1 (1970) · Zbl 0209.05503 [5] DOI: 10.1103/RevModPhys.57.617 · Zbl 0989.37516 · doi:10.1103/RevModPhys.57.617 [6] DOI: 10.1007/BF00386369 · Zbl 0805.58043 · doi:10.1007/BF00386369 [7] Dellnitz, The structure of symmetric attractors (1991) [8] Collet, Iterated Maps of the Interval as Dynamical Systems (1980) · Zbl 0458.58002 [9] DOI: 10.1016/0040-9383(76)90026-4 · Zbl 0346.58010 · doi:10.1016/0040-9383(76)90026-4 [10] Block, Ergod. Th. & Dynam. Sys. 6 pp 335– (1986) [11] Arrowsmith, An Introduction to Dynamical Systems (1990) · Zbl 0702.58002 [12] Mañé, Ergodic Theory and Differentiable Dynamics (1987) · doi:10.1007/978-3-642-70335-5 [13] MacKay, Hamiltonian Dynamical Systems (1987) · Zbl 0628.01045 [14] Katznelson, J. Anal. Math. 36 pp 156– (1979) [15] DOI: 10.1007/BF01394248 · Zbl 0475.58014 · doi:10.1007/BF01394248 [16] DOI: 10.1007/BF01300345 · Zbl 0676.54049 · doi:10.1007/BF01300345 [17] Hocking, Topology (1961) [18] Hirsch, Components of attractors (1992) [19] Guckenheimer, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2 [20] DOI: 10.1007/BF01982351 · Zbl 0429.58012 · doi:10.1007/BF01982351 [21] DOI: 10.1007/BF02096564 · Zbl 0771.58035 · doi:10.1007/BF02096564 [22] Willms, Ergod. Th. & Dynam. Sys. 8 pp 111– (1988) [23] Walters, An Introduction to Ergodic Theory (1982) · Zbl 0475.28009 · doi:10.1007/978-1-4612-5775-2 [24] Tang, J. Shanghai-Jiaotong-Univ. 125 pp 91– (1987) [25] DOI: 10.1090/S0002-9904-1967-11798-1 · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1 [26] Simmons, Introduction to Topology and Modern Analysis (1963) · Zbl 0105.30603 [27] Schweitzer, Ann. Math. 100 pp 368– (1974) [28] DOI: 10.1007/BF01212280 · Zbl 0595.58028 · doi:10.1007/BF01212280
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