The Yoccoz Combinatorial Analytic Invariant

In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz' proof of local connectivity of M at any Yoccoz parameter, i.e. any at most finitely renormalizable parameter for which all periodic orbits are repelling. Using this encoding we define an explicit combinatorial analytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic Mandelbrot set M₁ has the same combinatorial structure as M. As an immediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M.