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Communications in Mathematical Physics

, Volume 355, Issue 3, pp 1001–1119 | Cite as

Complex Bounds for Real Maps

  • Trevor Clark
  • Sebastian van Strien
  • Sofia Trejo
Open Access
Article
  • 374 Downloads

Abstract

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.

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Authors and Affiliations

  • Trevor Clark
    • 1
  • Sebastian van Strien
    • 1
  • Sofia Trejo
    • 1
  1. 1.Imperial College LondonLondonUK

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