Dilatations and exponents of quasisymmetric homeomorphisms (Q2790822)

A homeomorphism \(h\) of the real line \(\mathbb{R}\) onto itself is called quasisymmetric if and only if it is the boundary value of a quasiconformal mapping of the upper half plane \(\mathbb{H}\) onto itself. There are several conformal invariants to quantify the quasisymmetry of a homeomorphism. This paper gives a necessary and sufficient condition for a quasisymmetric homeomorphism \(h\) to have equal dilatation \(M_h\) and maximal dilatation \(K_h\). The dilatation of \(h\), denoted by \(M_h\), is defined as NEWLINE\[NEWLINEM_h=\sup_{A, B}\frac{\mathrm{Mod}(h(A),h(B);\mathbb{H})}{\mathrm{Mod}(A,B;\mathbb{H})},NEWLINE\]NEWLINE where the supremum is taken over all pairs of disjoint nondegenerate continua \(A\) and \(B\) on the real line. The maximal dilatation of \(h\), denoted by \(K_h\), is defined as NEWLINENEWLINE\[NEWLINEK_h=\inf\{K(f): f \text{ is a quasiconformal mapping of } \mathbb{H} \text{ onto itself with boundary value } h\}. NEWLINE\]NEWLINENEWLINEThere are examples showing that \(M_h\) and \(K_h\) are not equal to each other. If \(h\) is linear, then \(M_h=K_h=1\). So it is a natural and interesting problem to find out when the equality \(M_h=K_h\) holds.NEWLINENEWLINEThere is another important constant called the boundary dilatation of \(h\) defined by NEWLINENEWLINE\[NEWLINEH_h=\sup_{\zeta\in\mathbb{R}}H_h(\zeta),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEH_h(\zeta)=\inf\{K(f): f \text{ is a quasiconformal extension of } h \text{ in a neighborhood of }\zeta\in\mathbb{R}\}.NEWLINE\]NEWLINENEWLINEFor any quasisymmetric homeomorphism \(h\), \(H_h\leq K_h\) and \(M_h\leq K_h\). It was shown that if \(M_h=K_h\), then either \(H_h=K_h\) or \(h\) is induced by an affine mapping. Here ``\(h\) is induced by an affine mapping'' means that \(h=\phi_2\circ A_K\circ\phi_1^{-1}|_{\mathbb{R}}\), where \(\phi_1\) and \(\phi_2\) are conformal mappings between rectangles and \(\mathbb{H}\), and \(A_K(x+iy)=x+iKy\). NEWLINEHowever, it was known that this condition is not sufficient for \(M_h=K_h\), shown by some examples with \(H_h=K_h\) and \(M_h<K_h\). NEWLINENEWLINENEWLINENEWLINE In this paper, the authors introduce the quasisymmetric exponent of \(h\), denoted by \(\alpha_h\), and establish that \(M_h=K_h\) if and only if \(\alpha_h=K_h\) or \(h\) is induced by an affine mapping. The exponent is defined as \(\alpha_h=\sup_{x\in\mathbb{R}}\alpha_h(x)\), and \(\alpha_h(x)=\inf \lambda\), where for all distinct triples \(a,b,c\) in a neighborhood \(N\) of \(x\), if NEWLINE\[NEWLINE\frac{|c-b|}{|b-a|}\leq t,\;\;\text{ then} \;\;\frac{|h(c)-h(b)|}{|h(b)-h(a)|}\leq M\max\{t^{\lambda},t^{\frac{1}{\lambda}}\},NEWLINE\]NEWLINE where \(M\) is a constant. First, the authors compare these various conformal invariants and show that \(\alpha_h\leq M_h\leq K_h\) and \(\alpha_h\leq H_h\leq K_h\). In the proof of these inequalities, the Teichmüller function \(\Psi(t)\), \(t>0\), plays an important role. The Teichmüller function is defined byNEWLINE NEWLINE\[NEWLINE\mathrm{mod}([-1,0],[t,+\infty];\mathbb{C})=\frac{2\pi}{\ln\Psi(t)}NEWLINE\]NEWLINE NEWLINEand satisfies the well-known equality NEWLINE\[NEWLINE\lim_{t\rightarrow +\infty}\frac{\ln\Psi(t)}{\ln t}=1.NEWLINE\]NEWLINE Because of these inequalities, it is easy to see that \(\alpha_h=K_h\) is a stronger condition than \(H_h=K_h\), and it is obvious that if \(\alpha_h=K_h\) then \(M_h=K_h\). The next point is to show the necessary part, that is, if \(M_h=K_h\), then either \(h\) is induced by an affine mapping or \(\alpha_h=K_h\). In the proof, the authors analyze three cases of how \(M_h\) is attained. Apart from the cases where \(h\) is induced by an affine mapping, in the totally degenerate case, they get that \(M_h\leq \alpha_h\). So together with \(\alpha_h\leq M_h\) which is commonly true, if \(M_h=K_h\), then in the totally degenerate case, one has \(\alpha_h=M_h=K_h\). Using these invariants, a classification of the elements in the universal Teichmüller space is obtained in the paper, that is, \(\alpha_h=K_h\), \(\alpha_h<H_h=K_h\) (\(h\) has a substantial boundary point) or \(\alpha_h\leq H_h<K_h\) (\(h\) is a Strebel point).