Regular and limit sets for holomorphic correspondences (Q2717645)

A holomorphic correspondence is a multivalued map \(f\colon \widetilde{Q}_+\circ \widetilde{Q}_-:Z\to W\) where \(\widetilde{Q}_+\colon X\to W\), \(\widetilde{Q}_-\colon X\to Z\) are holomorphic surjective maps between Riemann surfaces. If \(Z=W\), such a correspondence can be iterated. The theory of iteration of holomorphic correspondences covers not only the iteration of rational maps, but also Kleinian groups, as seen by taking \(X\) the union of graphs of a set of generators and \(\widetilde{Q}_+\), \(\widetilde{Q}_-\) the projection maps.NEWLINENEWLINENEWLINEThere is a well-known parallel between iteration of rational maps and Kleinian groups which is illustrated by Sullivan's famous dictionary. In the paper under review, the foundations are laid for a third column to this dictionary which would correspond to holomorphic correspondences. Notions of regular set, limit set, normality and equicontinuity sets are defined. Contrary to the rational case, the last two do not coincide, and the Julia set is defined as the complement of the latter. The authors obtain invariance properties for these sets, and some relations between them, and state some conjectures, e.g. that the boundary of the limit set is a subset of the Julia set.NEWLINENEWLINENEWLINEThe authors also consider the difficult problem of constructing a fundamental domain for the iteration of a correspondence on its regular set, and solve it in a particular case.