Dynamics of certain smooth one-dimensional mappings I: The $C^{1+α}$-Denjoy-Koebe distortion lemma

We prove a technical lemma, the $C^{1+α}$-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a $C^{1+α}$ one-dimensional mapping $f:M\mapsto M$ with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.

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