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 <= 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.