Combinatorial configurations of fibered polynomials (Q2730718)

The object of the study are ``fibered polynomials'' of the form \(P\colon X\times{\mathbb C}\to X\times{\mathbb C}\), \((x,z)\mapsto (f(x),P_x(z)=z^2+c(x))\): this is the iteration of a quadratic polynomial \(z\mapsto z^2+c\) in which the parameter \(c\) depends on the iteration of the first coordinate, given by a continuous map \(f\colon X\to X\). This is a particular case of ``random iteration'' in which the function to be iterated varies in every interation step. The paper investigates such fibered quadratic polynomials from a combinatorial point of view, constructing a combinatorial model for (a subset of) the parameter space. This configuration space is proved to be a compact and connected Hausdorff space. Every nonrecurrent ``abstract configuration'' is shown to be realized by a unique quadratic fibered polynomial.