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Accessible saddles on fractal basin boundaries. (English) Zbl 0767.58023
From the authors’ summary: “For a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected and simply- connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods. Certain points on the basin boundary are distinguished by being accessible (by a path) from the interior of the basin. For an orientation-preserving homeomorphism, the accessible boundary points have a well-defined rotation number. Under some genericity assumptions, we prove that this rotation number is rational iff there are accessible periodic orbits. In particular, if the rotation number is the reduced fraction \(p/q\) and if the periodic orbits of periods \(q\) and smaller are isolated, then every accessible periodic orbit has minimum period \(q\). In addition, if the periodic orbits are hyperbolic, then every accessible point is on the stable manifold of an accessible periodic point.”
This paper is very interesting, well written and organized.
Reviewer: A.Klíč (Praha)

MSC:
37B99 Topological dynamics
30D40 Cluster sets, prime ends, boundary behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D99 Dynamical systems with hyperbolic behavior
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References:
[1] Bhatia, Stability Theory of Dynamical Systems (1970) · doi:10.1007/978-3-642-62006-5
[2] DOI: 10.1007/BF01456699 · JFM 44.0757.02 · doi:10.1007/BF01456699
[3] DOI: 10.1007/BF02100002 · Zbl 0746.58053 · doi:10.1007/BF02100002
[4] DOI: 10.1007/BF01223208 · Zbl 0729.58032 · doi:10.1007/BF01223208
[5] Mather, Adv. Math. Suppl Studies 7B pp none– (1981)
[6] DOI: 10.1007/BF02566283 · Zbl 0414.57002 · doi:10.1007/BF02566283
[7] Mather, Selected Studies pp 225– (1982)
[8] Levinson, Ann. Math. 50 pp 127– (1947)
[9] Hammel, Physica 35D pp 87– (1989)
[10] DOI: 10.1103/PhysRevLett.57.1284 · doi:10.1103/PhysRevLett.57.1284
[11] Dold, Lectures on Algebraic Topology (1972) · doi:10.1007/978-3-662-00756-3
[12] Grebogi, Physica none pp 243– (1987)
[13] Devaney, An Introduction to Chaotic Dynamical Systems (1986) · Zbl 0632.58005
[14] Collingwood, Cambridge Tracts in Mathematics and Mathematical Physics (1966)
[15] DOI: 10.2307/1969308 · doi:10.2307/1969308
[16] Birkhoff, Bull. Soc Math. 60 pp 1– (1932)
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