Rotation numbers of periodic orbits in the Hénon map
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. \textit{C. Grebogi}, \textit{E. Ott} and \textit{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.
scientific article; zbMATH DE number 4205449