The Mandelbrot set organizes the dynamics of polynomial maps. Polynomial-like maps — holomorphic maps that behave like polynomials on a neighborhood — carry copies of the Mandelbrot set in their parameter spaces, by the Douady-Hubbard straightening theorem. This is why the Mandelbrot set appears everywhere in polynomial dynamics.
Astorg, Benini, and Fagella build the transcendental analogue. Tangent-like maps are to meromorphic functions what polynomial-like maps are to polynomials — holomorphic maps with a single free asymptotic value that behave like the tangent family on a suitable domain. The model family has one parameter. Its bifurcation locus is the Tandelbrot set.
The straightening theorem carries over: every tangent-like map with connected filled Julia set is hybrid-equivalent to a unique member of the model family. The proof requires handling the asymptotic value — a fundamentally different type of singular value from a critical point — and establishing uniqueness through quasiconformal surgery adapted to the transcendental setting.
The parameter version follows: holomorphic families of tangent-like maps carry copies of the Tandelbrot set in their bifurcation loci, just as polynomial-like families carry copies of the Mandelbrot set.
The Mandelbrot set governs critical points. The Tandelbrot set governs asymptotic values. Together they organize the two types of singular values that control the dynamics of holomorphic maps. The taxonomy is now complete at the level of universal models: one set for each type of singularity.