Complex box bounds for real maps
File(s)1310.8338v3.pdf (1.6 MB) 10.1007%2Fs00220-017-2958-y.pdf (2.94 MB)
Accepted version
Published version
Author(s)
Clark, T
van Strien, S
Trejo, S
Type
Journal Article
Abstract
In this paper we prove complex bounds, also referred to as a priori bounds,
for real analytic (and even C3) interval maps. This means that we associate to
such a map a complex box mapping (which provides a kind of Markov structure),
together with universal geometric bounds on the shape of the domains. Such
bounds show that the first return maps to these domains are well-controlled,
and consequently form one of the corner stones in many recent results on
one-dimensional dynamics: renormalisation theory, rigidity results, density of
hyperbolicity, local connectivity.
for real analytic (and even C3) interval maps. This means that we associate to
such a map a complex box mapping (which provides a kind of Markov structure),
together with universal geometric bounds on the shape of the domains. Such
bounds show that the first return maps to these domains are well-controlled,
and consequently form one of the corner stones in many recent results on
one-dimensional dynamics: renormalisation theory, rigidity results, density of
hyperbolicity, local connectivity.
Date Issued
2017-11-01
Date Acceptance
2017-05-22
Citation
Communications in Mathematical Physics, 2017, 355 (3), pp.1001-1119
ISSN
0010-3616
Publisher
Springer
Start Page
1001
End Page
1119
Journal / Book Title
Communications in Mathematical Physics
Volume
355
Issue
3
Copyright Statement
© The Author(s) 2017
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
License URL
Sponsor
Commission of the European Communities
Identifier
http://arxiv.org/abs/1310.8338v3
Grant Number
339523
Subjects
math.DS
math.DS
37F10, 30D05
Publication Status
Published
Date Publish Online
2017-08-12