Complex box bounds for real maps

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.

References in corpus (3)

• Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials
• Absence of line fields and Mane's theorem for non-recurrent transcendental functions
• Monotonicity of entropy for real multimodal maps