2024/12/04 by Balázs Bárány, Bárány, Balázs, Károly Simon +3
Mathematics · #28A75 #28A80 #37E05 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2412.03529
openalex publication_date 2024/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a universal projection theorem, giving conditions on a parametrized family of maps Πλ: X → ℝd and a collection M of measures on X under which for almost every λ equality dimH Πλμ= min\d, dimH μ\ holds for all measures μ∈ M simultaneously (i.e. on a full measure set of λ's independent of μ). We require family Πλ to satisfy a transversality condition and collection M to satisfy a new condition called relative dimension separability. Under the same assumptions, we also prove that if the Assouad dimension of X is smaller than d, then for almost every λ, projection Πλ is nearly bi-Lipschitz (i.e. with pointwise α-Hölder inverse for every α∈ (0,1)) at μ-a.e. x, for all μ∈ M simultaneously. Our setting encompasses families of orthogonal projections, natural projections corresponding to conformal iterated function systems, and non-autonomous or random IFS. As applications, we provide novel results on the multifractal analysis, giving formula for the Hausdorff dimension of a level set of the local dimension for a typical (w.r.t the translation parameter) self-similar measure on the line, valid for the full range spectrum (including the decreasing part of the spectrum; previous results were covering only the increasing part). Among another applications, we prove that given a parametrized contracting conformal IFS satisfying the transversality condition, for almost every parameter the dimension formula holds for all ergodic shift-invariant measures simultaneously. We also prove that the dimension part of the Marstrand's projection theorem holds simultaneously for the collection of all ergodic measures on a strongly separated self-conformal set and for the collection of all Gibbs measures on a self-conformal set (without any separation).