The Thurston operator for semi-finite combinatorics

Given a continuous $l$-modal map $g$ of the interval $[0,1]$ we prove the existence of a polynomial $P$ with modality $\leq l$ such that $g$ is strongly semi-conjugate to $P$ in $[0,1]$. This is an improvement of a result in [4]. We do a modification on the Thurston operator in order to control the semi-finite combinatorial case. It turns out that all the essential attractors of $P$ have the same local topological type as those of $g$. This allows to construct the strong semi-conjugacy. We also present some examples agreeing with the results.