I will prove that for a hyperbolic meromorphic function f having a rapid derivative growth, if HD(J(f))>1, then the Jacobian of a probability invariant measure, equivalent to a conformal measure (with respect to a special metric associated to f and so-called tame geometric potential), has a real analytic extension on a neighbourhood of J(f)\f^–1(∞) in the complex plane. If, in addition, f satisfies a balanced derivative growth condition with constant exponents, then this extension is bounded in a neighbourhood of every pole of f.

Finally I will present a result, whose proof uses the real-analyticity of the Jacobian, i.e. the rigidity theorem for some class of meromorphic functions.