Hausdorff dimensions of quasilines varying in the universal Teichmüller space (Q2854219)

In this paper, the authors consider Hausdorff dimensions of quasilines. A homeomorphism \(f: \Omega\to \Omega'\) is called \textit{\(k\)-quasiconformal} if it satisfies the Beltrami equation NEWLINE\[NEWLINE \bar{\partial}f(z)=\mu(z)\partial f(z) \quad \text{a.e.\;\(z\in \Omega\)} NEWLINE\]NEWLINE with a measurable coefficient \(\mu\) with \(||\mu\||_\infty\leq k<1\). A \textit{\(k\)-quasiline} is the image of the real line under a \(k\)-quasiconformal mapping of the plane.NEWLINENEWLINE\textit{S. Smirnov} proved [Acta Math. 205, No. 1, 189--197 (2010; Zbl 1211.30037)] that the Hausdorff dimension of a \(k\)-quasiline is at most \(1+k^2\). \textit{K. Astala}'s conjecture [Acta Math. 173, No. 1, 37--60 (1994; Zbl 0815.30015)] (updated with Smirnov's result) says that for every \(0<k<1\), there exists a \(k\)-quasiline \(\alpha\) that has Hausdorff dimension \(\dim \alpha=1+k^2\).NEWLINENEWLINEThe universal Teichmüller space is the quotient space of all quasiconformal self-mappings of the upper half-plane that keep the three boundary points \(0\), \(1\), and \(\infty\) fixed, and two mappings are identified if they agree on the real line.NEWLINENEWLINELet me mention one result of the article, namely that there is an open and dense subset (Strebel points) of the universal Teichmüller space such that the corresponding quasilines have dimension strictly less than \(1+k^2\). The authors further show that there are some points \([f]\not=[\text{id}]\) outside above mentioned open and dense set in the universal Teichmüller space such that the Hausdorff dimension of the quasiline determined by \([f]\) is \(1\).