2012/11/01 by Christopher J. Bishop, Bishop, Christopher J., Hrant Hakobyan +3 · 1 citation
Computer Science · Decision Sciences · Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Fixed Point Theorems Analysis #Fuzzy and Soft Set Theory #Graph Labeling and Dimension Problems #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1211.0233
openalex publication_date 2012/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that if f:X→ Y is a quasisymmetric mapping between Ahlfors regular spaces, then dimH f(E)≤dimH E for "almost every" bounded Ahlfors regular set E⊆ X. If additionally, X and Y are Loewner spaces then dimH f(E)=dimH E for "almost every" Ahlfors regular set E⊂ X. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if f is a quasiconformal map of ℝN, N≥ 2, then for Lebesgue a.e. y∈ℝN we have dimH f(y+E) = dimH E. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E ⊂ ℝ is Ahlfors d-regular, d<1, then some component of f(E × ℝ) has dimension at most 2/(d+1), and we construct examples to show this bound is sharp. In addition, we show there is a 1-dimensional set S⊆ \mathbb R and planar quasiconformal map f such that f(ℝ × S) contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson \citeTyson:frequency and answer questions posed in \citeTyson:frequency and \citeAimPL.