An iterated implicit complex map
The authors' ultimate aim is to understand Hamiltonian systems, so they choose to study the iterations of a certain kind of complex mapping that at least possesses an analogue of time-symmetry. After a descriptive motivating survey, they gradually specialise to maps of the form \(z'=f(z)\) that solve an implicit equation \(g(z',z)=0\) where g is quadratic in each of z' and z. When the fixed points are forced to have suitable locations and multiplicities, the coefficients of g depend on just one arbitrary parameter, the rotation number \(\nu\). After this, the resulting map f is restricted to the upper half-plane, and its orbits and related phenomena are then discussed relative to the Julia-Fatou theory of complex maps. The analysis is partly empirical, though some assertions seem to have suppressed proofs - the style of writing does not clearly separate logical deduction from physical induction. The authors demonstrate that they have chosen an interesting type of mapping, but a satisfactory theory remains to be constructed (presumably by studying iterations on the entire complex curve \(g=0)\).
scientific article; zbMATH DE number 3975932