The focus of this book is on open conformal dynamical systems corresponding to the escape of a
point through an open Euclidean ball. The ultimate goal is to understand the asymptotic
behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic
conformal systems this has been addressed by various authors. The conformal maps considered in
this book are far more general and the analysis correspondingly more involved. The asymptotic
existence of escape rates is proved and they are calculated in the context of (finite or
infinite) countable alphabets uniformly contracting conformal graph-directed Markov systems
and in particular conformal countable alphabet iterated function systems. These results have
direct applications to interval maps rational functions and meromorphic maps. Towards this
goal the authors develop on a purely symbolic level a theory of singular perturbations of
Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite
type and Hölder continuous summable potentials. This leads to a fairly full account of the
structure of the corresponding open dynamical systems and their associated surviving sets.
