Regular and limit sets for holomorphic correspondences

S. Bullett¹ ; C. Penrose¹

¹School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London E1 4NS, UK

Fundamenta Mathematicae, Tome 167 (2001) no. 2, pp. 111-171

Résumé

Holomorphic correspondences are multivalued maps $f={\widetilde Q}_+{\widetilde Q}_-^{-1}:Z \rightarrow W$ between Riemann surfaces $Z$ and $W$, where ${\widetilde Q}_-$ and ${\widetilde Q}_+$ are (single-valued) holomorphic maps from another Riemann surface $X$ onto $Z$ and $W$ respectively. When $Z=W$ one can iterate $f$ forwards, backwards or globally (allowing arbitrarily many changes of direction from forwards to backwards and vice versa). Iterated holomorphic correspondences on the Riemann sphere display many of the features of the dynamics of Kleinian groups and rational maps, of which they are a generalization. We lay the foundations for a systematic study of regular and limit sets for holomorphic correspondences, and prove theorems concerning the structure of these sets applicable to large classes of such correspondences.

Détail

DOI : 10.4064/fm167-2-2

Keywords: holomorphic correspondences multivalued maps widetilde widetilde rightarrow between riemann surfaces where widetilde widetilde single valued holomorphic maps another riemann surface respectively iterate forwards backwards globally allowing arbitrarily many changes direction forwards backwards vice versa iterated holomorphic correspondences riemann sphere display many features dynamics kleinian groups rational maps which generalization lay foundations systematic study regular limit sets holomorphic correspondences prove theorems concerning structure these sets applicable large classes correspondences