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A Straightening Theorem for non-Autonomous Iteration

The classical straightening theorem as proved by Douady and Hubbard shows that a polynomial-like sequence is hybrid equivalent to a polynomial. We generalize this result to non-autonomous iteration where one considers composition sequences arising from a varying sequence of functions. In order to do this, new techniques are required to control the distortion and quasiconformal dilatation of the hybrid equivalence. In particular, the Caratheodory topology for pointed domains allows us to specify the appropriate bounds on the sequence of sets on which the polynomial-like mapping sequence is defined and give us good estimates on the degree of distortion and quasiconformality.