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 using the “toy example” of the exponential family.