Distortion of parallel lines under spatial quasiconformal mappings

Problem's Description

Is there a subset $S\subset \mathbb{R}^2$ such that $\dim_H S > 1$ and a quasiconformal map $f : \mathbb{R}^n\to\mathbb{R}^n, n\geq 3$, such that $f(\{x\}\times (0,1))$ contains no nontrivial rectifiable arc for any $x\in S$?

1. ArticleIs an originQuasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

   • Christopher J. Bishop,
   • Hrant Hakobyan,
   • Marshall Williams

   Geometric And Functional Analysis 26 (2), 379-421, 2016

2. ArticleIs an originFrequency of Sobolev and quasiconformal dimension distortion

   • Zoltan M. Balogh,
   • Roberto Monti,
   • Jeremy T Tyson

   Journal de Mathématiques Pures et Appliquées 99 (2), 125-149, 2013