Complex box bounds for real maps

Trevor Clark; Sebastian van Strien; Sofia Trejo

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arXiv · 1310.8338

Complex box bounds for real maps

Trevor Clark·Sebastian van Strien·Sofia Trejo

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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.

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Trevor Clark, Sebastian van Strien, Sofia Trejo. 2013-10-30. Complex box bounds for real maps. https://arxiv.org/abs/1310.8338

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