通过Herglotz测度与Koenigs线性化刻画极值双曲速率
Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization
AI总结：
本文利用Herglotz测度与Koenigs线性化，刻画单位圆盘双曲全纯自映射的极值双曲速率，将理论推广至多连通域、单位球及拟共形映射，明确其增长由Denjoy-Wolff点处边界线性化非退化性决定。
AI中文摘要：
设$g:\mathbb{D}\to\mathbb{D}$是单位圆盘的双曲全纯自映射，其Denjoy-Wolff点为$\tau\in\partial\mathbb{D}$，角导数为$\alpha\in(0,1)$。若对任意$z,w\in\mathbb{D}$，其向前迭代满足尖锐度量渐近$\rho_{\mathbb{D}}(g^{\circ n}(z),w) = n\log\frac{1}{\alpha} + O(1)$（当$n\to\infty$时），则称$g$具有\"极值双曲速率\"。利用Herglotz-Nevanlinna表示，本文证明：$g$具有极值双曲速率当且仅当关联的边界测度$\sigma$满足$\int_{\partial\mathbb{D}\setminus\{\tau\}} \log\frac{1}{|\zeta-\tau|} \\,d\sigma(\zeta) < \infty$。进一步证明该条件等价于Koenigs线性化的非退化角渐近：对任意满足$\phi_\tau(\tau)=\infty$的共形映射$\phi_\tau:\mathbb{D}\to\mathbb{H}$，有$0< \left| \angle\lim_{z\to\tau} \frac{h(z)}{\phi_\tau(z)} \right| < \infty$，其中$h$为Koenigs函数。本文将极值速率理论推广至单位圆盘之外：对有限连通双曲平面域，圆盘上的刻画可推广至关联甲板变换群的普通边界点；对单位球$\mathbb{B}^n$的全纯自映射及$K$-拟共形自映射（此类映射无可比的Herglotz表示），本文建立了保证极值双曲速率的充分边界正则条件，并将拟共形结果进一步推广至有限连通平面域的普通边界点。上述结果表明，在所考虑的各类情形中，极值双曲增长由Denjoy-Wolff点处边界线性化的非退化性决定。
英文摘要：
Let $g:\mathbb{D}\to\mathbb{D}$ be a hyperbolic holomorphic self-map of the unit disk with Denjoy--Wolff point $τ\in\partial\mathbb{D}$ and angular derivative $α\in(0,1)$. We say that $g$ has an \textit{extremal hyperbolic rate} if its forward iterates satisfy the sharp metric asymptotic \begin{equation*} ρ_{\mathbb{D}}(g^{\circ n}(z),w) = n\log\frac{1}α + O(1) \quad\text{as }n\to\infty, \end{equation*} for every $z,w\in\mathbb{D}$. Using the Herglotz--Nevanlinna representation, we prove that $g$ has an extremal hyperbolic rate if and only if the associated boundary measure $σ$ satisfies \begin{equation*} \int_{\partial\mathbb{D}\setminus\{τ\}} \log\frac{1}{|ζ-τ|} \,dσ(ζ) < \infty. \end{equation*} We further show that this condition is equivalent to a non-degenerate angular asymptotic of the Koenigs linearization: for any conformal map $ϕ_τ:\mathbb{D}\to\mathbb{H}$ with $ϕ_τ(τ)=\infty$, \begin{equation*} 0< \left| \angle\lim_{z\toτ} \frac{h(z)}{ϕ_τ(z)} \right| < \infty, \end{equation*} where $h$ is a Koenigs function. We then extend the extremal-rate theory beyond the unit disk. For finitely connected hyperbolic planar domains, the disk characterizations transfer to ordinary boundary points of the associated deck transformation group. For holomorphic self-maps of the unit ball $\mathbb{B}^n$ and for $K$-quasiconformal self-maps, where no comparable Herglotz representation is available; we establish sufficient boundary regularity conditions that guarantee the extremal hyperbolic rate. The quasiconformal result is further extended to finitely connected planar domains at ordinary boundary points. These results show that, across the settings considered here, extremal hyperbolic growth is governed by the non-degeneracy of the boundary linearization at the Denjoy--Wolff point.
