2004/08/21 by Artur Avila, Avila, Artur, Mikhail Lyubich +1
Computer Science · Mathematics · #37F35 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.DS #msc:37F35
paper · pdf · doi:10.48550/arxiv.math/0408290
Latex, 51 pages
arxiv created 2004/08/21 · openalex publication_date 2004/08/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets J(f) whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent \de_\crit is equal to the hyperbolic dimension \HD_\hyp(J(f)). Moreover, if \area J(f)=0 then \HD_\hyp (J(f))=\HD(J(f)). In the stationary case, the last statement can be reversed: if \area J(f)> 0 then \HD_\hyp (J(f))< 2. We also give a new construction of conformal measures on J(f) that implies that they exist for any \de∈ [\de_\crit, ∞), and analyze their scaling and dissipativity/conservativity properties.