Complex Bounds for Real Maps

Complex Bounds for Real Maps In this paper we prove complex bounds, also referred to as a priori bounds for C 3, and, in particular, for analytic maps of the interval. Any C 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval. We associate to such a map, a complex box mapping, which provides a kind of Markov structure for the dynamics. Moreover, we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain. Such bounds show that the first return maps to these domains are well controlled, and consequently such bounds form one of the corner stones in many recent results in one-dimensional dynamics, for example: renormalization theory, rigidity, density of hyperbolicity, and local connectivity of Julia sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by The Author(s)
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-2958-y
Publisher site
See Article on Publisher Site

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