2011/04/07 by Julien Barral, Barral, Julien, Yan-Hui Qu +1
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary: 37B40 #Quantum chaos and dynamical systems #Secondary: 28A80 #math.DS #msc:28A80 #msc:37B40
paper · pdf · doi:10.48550/arxiv.1104.1442
arxiv created 2011/04/07 · openalex publication_date 2011/04/07 · arxiv updated 2011/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials (ϕn)n=1^∞ on a topologically mixing subshift of finite type X endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of \it weak concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of ϕn(x) is localized, i.e. depends on the point x rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form \x∈ X: limn→∞ ϕn(x)/n=ξ(x)\, where ξ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in \Rd, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.