Analytic mappings of the unit disk which almost preserve hyperbolic area (Q6639750)

Let \(\mathbb{D}\) be the open unit disk in \(\mathbb{C}\) and let \(A_h(E)\) be the hyperbolic area of a measurable set \(E\subset\mathbb{D}\). Let \(B_h(z,R)\) be the hyperbolic disk centered at \(z\in \mathbb{D}\) of hyperbolic radius \(R>0\). It is said that an analytic self-mapping \(F\) of \(\mathbb{D}\) almost preserves the hyperbolic area (APHA) if there exists a constant \(c>0\) such that \N\[\NA_h(F(B_h(z,R)))\ge c\,A_h(B_h(z,R)),\quad z\in\mathbb{D},\quad R>1. \N\]\NThe Lyapunov exponent of a Borel probability measure \(\sigma\) on \(\partial\mathbb{D}\) is defined as \N\[\N\chi(\sigma,F)=\int_{\partial\mathbb{D}}\log|F'|d\sigma. \N\]\NFor each point \(\alpha\in\partial{\mathbb{D}}\), the Aleksandrov-Clark measure of \(F\) is the positive measure \(\sigma_\alpha\) on \(\partial{\mathbb{D}}\) such that for a constant \(C_\alpha\in\mathbb{R}\)\N\[\N\dfrac{\alpha +F(z)}{\alpha-F(z)}=\int_{\partial{\mathbb{D}}}\dfrac{\xi+z}{\xi-z}d\sigma_\alpha(\xi)+iC_\alpha,\quad z\in\mathbb{D}. \N\]\NIn this very interesting paper, the authors provide several characterizations of APHA mappings through results that involve notions such as almost isometry, finite angular derivative, finite entropy, Mobius distortion, maximal Blaschke product, Carleson-Newman Blaschke product, Lyapunov exponent, Lipschitz function, among others. They describe Aleksandrov-Clark measures \(\sigma\) of APHA mappings as those for which the function \(H[\sigma]\) defined by \N\[\NH[\sigma](z)=\int_{\partial\mathbb{D}}\dfrac{d\sigma(\xi)}{|\xi-z|^2},\quad z\in\overline{\mathbb{D}}, \N\]\Nverifies \(\log H[\sigma]\in\) BMO(\(\partial\mathbb{D}\)) and there exists a constant \(C>0\) so that \N\[\N\log H[\sigma](z_I)-C\le \frac{1}{m(I)}\int_I \log H[\sigma](\xi)dm(\xi)\le \log H[\sigma](z_I)+C, \N\]\Nfor any arc \(I\subset \partial\mathbb{D}\) (\(z_I=(1-m(I))\xi\)). The authors also carry out an extensive study of the Lyapunov exponents of Aleksandrov-Clark measures of APHA mappings. In particular, they prove a crucial result: If \(F\) is an APHA mapping and \(\{\sigma_\alpha:\alpha\in\partial\mathbb{D}\}\) is the collection of its Aleksandrov-Clark measures, then \(\chi(\sigma_\alpha,F)\) agrees with a \(C^\infty\) function a.e. on \(\partial\mathbb{D}\).