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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Perturbations of weakly expanding critical orbits

Perturbations of weakly expanding critical orbits

(p.163) Perturbations of weakly expanding critical orbits
Frontiers in Complex Dynamics

Genadi Levin

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Princeton University Press

This chapter studies perturbations of polynomials and rational functions with several (possibly, not all) summable critical points. It proves that there exists an r-dimensional manifold Δ‎ in an appropriate space containing f (a polynomial or a rational function which has r summable critical points) such that for every smooth curve in Δ‎ through f, the ratio between parameter and dynamical derivatives along forward iterates of at least one of these summable points tends to a non-zero number. In doing so the chapter establishes a precise form of this relation for rational maps with one critical point satisfying the summability condition (certain expansion rate assumption along the critical orbit). This result brings to a natural general form many previously known special cases studied over the years.

Keywords:   summability condition, perturbations, expanding critical orbits, summable critical points, rational maps, dynamical derivatives

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