A straightening theorem for non-autonomous iteration (Q2906307)

The straightening theorem due to \textit{A. Douady} and \textit{J. H. Hubbard} states that every polynomial-like mapping \(f: U'\to U\) of degree \(d\) is hybrid equivalent to a polynomial \(P\) of degree \(d\) [Ann. Sci. Éc. Norm. Supér. (4) 18, 287--343 (1985; Zbl 0587.30028)].NEWLINENEWLINE In this paper, the author extends this result to the setting of non-autonomous polynomial iteration, where one composes a sequence of mappings which in general are allowed to vary. The non-autonomous version of the straightening theorem is then the following:NEWLINENEWLINE Let \((\mathcal F: \mathcal{U}\to\mathcal{V})\) be a polynomial-like mapping sequence with degree bounded by \(d\). Then there exists a polynomial-like restriction \((\mathcal F: \mathcal{U}'\to\mathcal{V}')\) and a bounded monic centered sequence of polynomials \(\{P_m\}_{m=1}^{\infty}\) such that \((\mathcal F: \mathcal{U}'\to\mathcal{V}')\) is hybrid equivalent to \(\{P_m\}_{m=1}^{\infty}\). Moreover, the degree of each \(P_{m+1}\) is \(d_{m+1}\), the degree of \(f_{m+1}\).NEWLINENEWLINE For the proof, the author uses the Carathéodory topology for pointed domains.