Invariant measures for meromorphic Misiurewicz maps

Invariant measures for meromorphic Misiurewicz maps

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 conditio...

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Veröffentlicht in:Mathematical proceedings of the Cambridge Philosophical Society 2008-11, Vol.145 (3), p.685-697
Hauptverfasser: KOTUS, JANINA, ŚWIATEK, GRZEGORZ
Format: Artikel
Sprache:eng
Schlagworte:
Mathematics
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Zusammenfassung: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 on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004108001412