2013/10/30 by Trevor Clark, Clark, Trevor, Sebastian van Strien +3 · 1 citation
Mathematics · #30D05 #37F10 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:30D05 #msc:37F10
paper · pdf · doi:10.48550/arxiv.1310.8338
arxiv created 2017/01/05 · arxiv updated 2017/01/06
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.