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The Mandelbrot set is a fractal object encoding the dynamical behaviour of the family of quadratic polynomials z^2+c, where c is a parameter varying over the complex plane. It surprisingly appears also in the parameter spaces of all (reasonable) rational maps and in such sense, it also encodes the dynamical behaviour of this much larger class. The explanation is intricate and relies on the concept of renormalization: essentially, renormalization isolates and extrapolates the behaviour of a rational functions near its critical values, and brings it back to analogous behaviours for quadratic polynomials. In this work we present an analogous object for transcendental maps, which arises from a model family and yet encodes the dynamical behaviour of all (reasonable) families of transcendental meromorphic maps.
This is joint work with M. Astorg and N. Fagella.