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A model for the Mandelbrot set is due to Thurston and is stated in thelanguage of geodesic laminations. The conjecture that the Mandelbrot set isactually homeomorphic to this model is equivalent to the celebrated MLCconjecture stating that the Mandelbrot set is locally connected. For parameterspaces of higher degree polynomials, even conjectural models are missing, onepossible reason being that the higher degree analog of the MLC conjecture isknown to be false. We provide a combinatorial model for an essential part ofthe parameter space of complex cubic polynomials, namely, for the space of allcubic polynomials with connected Julia sets all of whose cycles are repelling(we call such polynomials \emph{dendritic}). The description of the model turnsout to be very similar to that of Thurston.

Combinatorial models for spaces of cubic polynomials