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Rotation numbers of periodic orbits in the Hénon map. (English) Zbl 0729.58032
It is known that for an area-contracting diffeomorphism f: \({\mathbb{R}}^ 2\to {\mathbb{R}}^ 2\) with more than one attractor, the boundaries between respective basins of attraction can be extremely complicated, e.g. can be fractal and contain infinitely many unstable periodic orbits. C. Grebogi, E. Ott and J. Yorke [Physica D 24, 243-262 (1987; Zbl 0613.58018)] have distinguished certain orbits of the basin boundary by being accessible by a path from the interior of the basin. For connected and simply connected basins of attraction the diffeomorphism f acts on the set S of accessible points as if they were on a circle, and so one can associate a rotation number. Although in a one-parameter family of invertible circle maps the rotation number varies continuously with the parameter, this factor is not valid for f/S. The purpose of the paper under review is to study changes in the accessible rotation numbers as map parameters vary. The main result is the following.
Theorem. Let \(f_{\lambda}\) be a \(C^ 3\)-family of real analytic, area- contracting orientation-preserving maps of the plane. Suppose that a periodic saddle p forms a rotary homoclinic tangency at a point \(q\in orb(p)\) when \(\lambda =\lambda_*\) (i.e. there is a tangency of the unstable manifold of q with the stable manifold of an adjacent point from orb(p)). Then there exists a sequence \(\lambda_ n\to \lambda_*\) and a sequence of saddles \(p_ n\to q\) such that each saddle \(p_ n\) has a rotary homoclinic tangency at \(\lambda_ n.\)
A numerical investigation of the Henon map as a particular case is given, too.

37B99 Topological dynamics
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
Full Text: DOI
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