Periodic points of holomorphic twist maps (Q2574741)

Let us consider a holomorphic twist map \[ H(z,w)=\left(\lambda ze^{\varphi(zw)} + O\biggl(\bigl\|(z,w)\bigr\|^{2n+2}\biggr),\;\overline \lambda we^{-\varphi(zw)}+O \biggl(\bigl\|(z,w)\bigr\|^{2n+2}\biggr) \right) \] defined on a neighborhood of the origin in \(\mathbb{C}^2\), where \(\varphi\) is a holomorphic function vanishing to order \(n\) at the origin and where \(H^*(dz\wedge dw)=dz\wedge dw\), i.e., \(H\) is symplectic and \(\lambda\) is a complex number of modulus 1. Consider the following problem: Does there exist a sequence of periodic orbits \(\{p_{n,1}, p_{n,2}=H(p_{n,1}),\dots,p_{n,k_n+1}=H^{k_n}(p_{n,1})= p_{n,1} \}_n\) converging to the origin? It is known that the answer to the problem is affirmative in the case when \(H(z,w)\) is a reversible mapping. In the symplectic case it remains open. In this paper, the author investigates what happens to periodic orbits when the map is perturbed to the form \(H_{\varepsilon(z,w)}=(\lambda ze^{\varphi (zw)}+\varepsilon\psi (\overline\lambda we^{-\varphi(zw)}),\overline\lambda we^{-\varphi (zw)})\) with some hypotheses on \(\varphi,\psi\). His main result is: For any integer \(k\geq 1\) there is an \(\varepsilon(k)>0\) so that for any \(\varepsilon \in\mathbb{C}\), \(0<|\varepsilon|<\varepsilon(k)\), the map \(H\) has no periodic orbits in the unit ball with period \(k\) (except the origin). The interesting cases of Hénon maps and holomorphic Hamiltonian flows are discussed at the end of the paper.