In complex dynamics it is often important to understand and
describe the dynamical behavior of critical (or singular) orbits. For
quadratic polynomials, this leads to the study of the Mandelbrot set and
of its complement. In our talk we present a theorem which classifies
within certain families the transcendental entire functions for which
all singular values escape, that is, inside of the complement of the
"transcendental analogue" of the Mandelbrot set. A key to the proof of
the theorem is a generalization of the celebrated Thurston's Topological
Characterization of Rational Functions, but for the case of infinite
rather than finite set of "punctures". As in the classical theorem of
Thurston, we consider a special "sigma-map" acting on a Teichmüller
space which is in our case infinite-dimensional.
We give a brief overview of the project, and afterwards we discuss some
of the main ingredients involved, such as e.g. "spiders" with
infinitely many "legs".