# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Regular and limit sets for holomorphic correspondences

### Tom 167 / 2001

Fundamenta Mathematicae 167 (2001), 111-171 MSC: Primary 37F05; Secondary 30D05, 30F40. DOI: 10.4064/fm167-2-2

#### Streszczenie

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.

#### Autorzy

• S. BullettSchool of Mathematical Sciences
Queen Mary and Westfield College
University of London
London E1 4NS, UK
e-mail
• C. PenroseSchool of Mathematical Sciences
Queen Mary and Westfield College
University of London