Invariant measures for meromorphic Misiurewicz maps

摘要

We study the existence of finite absolutely Continuous invariant measures for meromorphic Misiurewicz maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that Such a measure Must exist. We show that it is not so for meromorphic maps unless an additional condition oil the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.