In holomorphic dynamics it is often important to understand
the dynamical behaviour of critical (or singular) orbits. For quadratic
polynomials, this leads to the study of the Mandelbrot set and of its
complement. We present a classification of some explicit families of the
transcendental entire functions for which all singular values escape,
i.e. of the functions belonging to the complement of the "transcendental
analogue" of the Mandelbrot set. A key ingredient in its proof is a
generalisation of the famous Thurston's Topological Characterization of
Rational Functions, but for the case of infinite rather than finite
post-singular set. Analogously to Thurston's theorem, we consider a
dynamical system in a specially chosen Teichmüller space and investigate
its convergence. Unlike the classical case, the underlying Teichmüller
space is infinite-dimensional which leads to completely different
constructions.

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