PROBABILISTIC INVARIANT MEASURES FOR NON-ENTIRE FUNCTIONS WITH ASYMPTOTIC VALUES MAPPED ONTO ∞

摘要

We study the dynamics of transcendental meromorphic functions of the form f(z) = R ο exp(z), where R is a non-constant rational map and both asymptotic values R(0) and R(∞) are eventually mapped onto ∞. With each map f we associate its projection F on the cylinder P. Let J_F~r consist of all points whose trajectory returns infinitely often to some compact set whose intersection with the postsingular set is empty, and let h = HD(J_F~r) be the Hausdorff dimension of J_F~r. We prove that the h-dimensional Hausdorff measure H~h of J_F~r is positive and finite, while the h-dimensional packing measure of J_F~r is locally infinite at every point of this set. We also prove that there exists a unique F-invariant Borel probability measure μ on J_F~r that is absolutely continuous with respect to the Hausdorff measure H~h, and that μ is ergodic and conservative.