A note on a modified version of the Poincaré-Birkhoff theorem
Consider the universal covering of the plane \({\mathcal C}=\{(\theta,r):\theta\in{\mathbb R}\), \(r\geq 0\}\). The authors prove the existence of an area-preserving homeomorphism \(\widetilde\psi:{\mathcal C}\to{\mathcal C}\) devoid of fixed points in the interior of \({\mathcal C}\) and such that \(\widetilde\psi(\theta,r)=(\theta+\Theta(\theta,r),{\mathcal R}(\theta,r))\), where \({\mathcal R}\) and \(\Theta\) are \(2\pi\)-periodic in \(\theta\), \({\mathcal R}(\theta,0)=0\) \(\forall\theta\in{\mathbb R}\), and satisfy (1) \(\exists \widetilde\theta\) s.t.\ \(\limsup_{r\to+\infty}\Theta(\widetilde\theta,r)<0\), (2) \(\Theta(\theta,0)>0\), \(\forall\theta\in {\mathbb R}\). This is particularly significant as it shows that the ``twist'' conditions at zero and at infinity in the following asymptotic version of the Poincaré-Birkhoff theorem cannot be interchanged (Asymptotic P-B theorem): Let \(\psi:{\mathcal C}\to{\mathcal C}\) be an area-preserving homeomorphism such that \(\psi(\theta,r)=(\theta+\Theta(\theta,r),{\mathcal R}(\theta,r))\), where \({\mathcal R}\) and \(\Theta\) are \(2\pi\)-periodic in \(\theta\), \({\mathcal R}(\theta,0)=0\) \(\forall\theta\in{\mathbb R}\). Assume also that the following conditions hold: (i\(_\infty\)) \(\forall\theta\in {\mathbb R}\) \(\liminf_{r\to+\infty}\Theta(\theta,r)>0\) (or \(\limsup_{r\to+\infty}\Theta(\theta,r)<0\)), (ii) \(\exists\overline\theta\in{\mathbb R}\) s.t.\ \(\Theta(\overline\theta,0)<0\) (respectively \(\Theta(\overline\theta,0)>0\)). Then, \(\psi\) admits at least one fixed point \((\theta^*,r^*)\) in the interior of \({\mathcal C}\) with \(0\leq\theta^*<2\pi\). If such a fixed point is unique then its fixed point index is nonzero. Considering the restriction of \(\widetilde\psi\) to the stripe \({\mathbb R}\times [0,R]\) for sufficiently large \(R\), one can conclude that the ``twist'' conditions on the curves \(\Gamma_1\) and \(\Gamma_2\) cannot be interchanged also in the following version of the Poincaré-Birkhoff theorem due to \textit{A. Margheri, C. Rebelo} and \textit{F. Zanolin} [J. Differ. Equations 183, 342--367 (2002; Zbl 1119.37323)] (that implies the above asymptotic version) (Modified P-B theorem): Let \(\psi:{\mathcal C}\to{\mathcal C}\) be an area-preserving homeomorphism such that \(\psi(\theta,r)=(\theta+\Theta(\theta,r),{\mathcal R}(\theta,r))\), where \({\mathcal R}\) and \(\Theta\) are \(2\pi\)-periodic in \(\theta\), \({\mathcal R}(\theta,0)=0\) \(\forall\theta\in{\mathbb R}\). Take \(\Gamma_1=\{(\theta,r):r=0\), \(\theta\in{\mathbb R}\}\) and \(\Gamma_2=\{(\theta,r):r=\beta(\theta)\), \(\theta\in{\mathbb R}\}\) where \(\beta\) is a continuous \(2\pi\)-periodic positive function. Assume also that the following conditions hold: (i) \(\Theta| _{\Gamma_2}>0\) (or \(\Theta| _{\Gamma_2}<0\)), (ii) \(\exists(\overline\theta,0)\in\Gamma_1\) s.t.\ \(\Theta(\overline\theta,0)<0\) (respectively \(\Theta(\overline\theta,0)>0\)). Then, \(\psi\) admits at least one fixed point \((\theta^*,r^*)\) in the interior of the stripe \(\{(\theta,r):\theta\in{\mathbb R},0\leq r\leq\beta(\theta)\}\) with \(0\leq\theta^*<2\pi\). If such a fixed point is unique then its fixed point index is nonzero. The authors, as a further remark, also provide a simpler construction, valid for the situation relative to the latter theorem, showing the noninterchangeability of the ``twist'' conditions in this case.
scientific article; zbMATH DE number 2093538