Self-rotation number using the turning angle (Q1586153)

For circle homeomorphisms or monotone twist area preserving maps on the annulus \(\mathbb{S}^1\times\mathbb{R}\), the notion of rotation number of an orbit is well known. If the map \(F\) is defined on \(\mathbb{R}^2\), and \(x^*\) is a fixed point for \(F\), then we can define the rotation number of the orbit \({\mathcal O}(x)\) as the rotation number of the point \(x\) associated to the map \(F\) restricted to \(\mathbb{R}^2\setminus\{x^*\}\cong\mathbb{S}^1\times\mathbb{R}^+\). In this case the rotation number measures the average rate of rotation around \(x^{*}\). This definition has a drawback: the rotation number can change discontinuously as the parameter in the map's definition varies continuously, even when no bifurcations occur. In [\textit{B. B. Peckham}, Nonlinearity 3, No. 2, 261-280 (1990; Zbl 0704.58035)] the self-rotation number of a periodic orbit of an homeomorphism of \(\mathbb{R}^{2}\) is defined, as an alternative notion of rotation number. In order to simplify the computation of the self-rotation number, the paper under review proposes a construction of this number based on the so called turning angle. The turning angle of an orbit of the map \(F\) measures the average rate of turning of the vector \(V=F(y)-y\) along the orbit \({\mathcal O}(x)\). Let \(\varphi:\mathbb{R}^{2}\to\mathbb{S}^{1}\) be the function that associates to a point \(x\in\mathbb{R}^{2}\) its turning angle \(\theta\in\mathbb{S}^{1}\). Then the self-rotation number of \(x\) is defined by: \[ \rho_s(x)=\lim_{n\to\infty}\sum_{i=1}^{n}\varphi(F^{i}(x)) \] The function \(\varphi\) is not surjective in general. If \(\alpha\not\in\text{Im}(\varphi)\), we say that the homeomorphism \(F\) avoids the angle \(\alpha\). It is proved that if a one parameter family \(f_\mu\) of the plane avoids the angle \(\alpha(\mu)\), then the self-rotation number \(\rho_{s}(x)\) is invariant under continuation of periodic orbits of \(f_\mu\). For example every orientation preserving nonsingular linear map \(L:\mathbb{R}^{2}\to\mathbb{R}^{2}\) avoids an angle \(\alpha\). As a consequence, every near linear map avoids an angle. The orientation preserving Hénon map avoids the angle \(\alpha=-3\pi/2\), as well as a class of maps, called Hénon-like maps. The authors show how to compute the self-rotation number for orbits of the Hénon map at the integrable limit. Thus the self-rotation number for a symbolic sequence is defined.