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The study of escape rates for a ball in a dynamical systems has been muchstudied. Understanding the asymptotic behavior of the escape rate as the radiusof the ball tends to zero is an especially subtle problem. In the case ofhyperbolic conformal systems this has been addressed by various authors. Inthis paper we consider a far more general realm of conformal maps where theanalysis is correspondingly more complicated. We prove the existence of escaperates and calculate them in the context of countable alphabets, either finiteor infinite, uniformly contracting conformal graph directed Markov systems withtheir special case of conformal countable alphabet iterated function systems.This goal is achieved by developing the appropriate theory of singularperturbations of Perron-Frobenius (transfer) operators associated withcountable alphabet subshifts of finite type and H\"older continuous summablepotentials. This is the key ingredient for further results about otherconformal systems. These include topological Collet-Eckmann multimodal intervalmaps and rational maps of the Riemann sphere (an equivalent formulation is tobe uniformly hyperbolic on periodic points), and also a large class oftranscendental meromorphic functions.

Non-Escaping Sets in Conformal Dynamical Systems and Singular Perturbations of Perron-Frobenius Operators