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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

Chapter:
(p.163) Perturbations of weakly expanding critical orbits
Source:
Frontiers in Complex Dynamics
Author(s):

Genadi Levin

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0010

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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