Holomorphic symplectomorphisms in \(\mathbb{C}^{2p}\) (Q1919626)

The authors study the properties of biholomorphic symplectomorphisms of \(\mathbb{C}^{2p}\), mainly the structure of the space of periodic or quasi-periodic orbits. They show that for the time-independent holomorphic Hamiltonians in \(\mathbb{C}^{2p}\), generically all orbits of the corresponding biholomorphic symplectomorphisms go to infinity. They also confirm Herman's conjecture, i.e. that there is a \(G_\delta\) dense set \(S'\) of the space of biholomorphic symplectomorphisms of \(\mathbb{C}^{2p}\) (endowed with the topology of uniform convergence on compact sets) such that for \(f\in S'\) the set \(\{(z, w);\;f^n(z,w)\) is bounded\} has empty interior.