Communications in Mathematical Physics

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

Complex Bounds for Real Maps

  • Trevor Clark
  • Sebastian van StrienEmail author
  • Sofia Trejo
Open Access


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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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

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

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