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In complex dynamics, we construct a so-called nice set (one for which thefirst return map is Markov) around any point which is in the Julia set but notin the post-singular set, adapting a construction of Juan Rivera-Letelier. Thissimplifies the study of absolutely continuous invariant measures. We prove aconverse to a recent theorem of Kotus and Swiatek, so for a certain class ofmeromorphic maps the absolutely continuous invariant measure is finite if andonly if an integrability condition is satisfied.

Nice sets and invariant densities in complex dynamics