Absolutely Continuous Invariant Measures for Expansive Rational Maps with Rationally Indifferent Periodic Points

Expansive rational maps T:C→ C which are not expanding with respect to the spherical metric are those which have rationally indifferent periodic points. For an atomless t-conformal measure m of such a rational map we prove the existence of a unique (up to a multiplicative constant) σ-finite, T-invariant measureμabsolutely continuous with respect to m. We also give a necessary and sufficient condition for the measureμ to be finite.