Fractal straight lines and quasisymmetries (Q1920876)

The author continues his studies [\textit{D. A. Trotsenko}: Sib. Mat. Zh. 27, No. 6 (160), 196-205 (1986; Zbl 0635.30022)] and Sib. Mat. Zh. 28, No. 6(166), 126-133 (1987; Zbl 0656.30015)] of quasisymmetric maps which are close to similarities. This paper is devoted to curves which are almost straight lines. A curve \(\gamma : \overline \mathbb{R} \to \mathbb{R}^n\), \(\gamma (\infty) = \infty\), satisfies condition \(J (\varepsilon)\) if for all \(x,y \in R\), \(\text{dist} (\gamma [x,y], [\gamma (x), \gamma (y)] \leq \varepsilon |\gamma (x) - \gamma (y) |\). Here \(\text{dist} (A,B) = \sup \{d (x,B) : x \in A\}\) and \(d\) stands for the euclidean distance. For \(h > 0\) a mapping \(f : \mathbb{R} \to \mathbb{R}^n\) is called an \(h\)-similarity if for each \(x_0 \in \mathbb{R}\) and \(r > 0\) there is a similarity \(T : \mathbb{R} \to \mathbb{R}^n\) such that \(|T (f(x)) - x |\leq hr\) for all \(x \in (x_0 - r,x_0 + r)\). The main theorem says that if a curve \(\gamma\) satisfies condition \(J (\varepsilon)\) for small, explicit \(\varepsilon > 0\), then there is an \(h\)-similarity \(f : \mathbb{R} \to \mathbb{R}^n\) with \(f(R) = \gamma (R)\) and \(h \leq 1500 \varepsilon\). The proof is very technical.