Symmetrization and the rate of convergence of semigroups of holomorphic functions

TL;DR

The paper shows that Steiner symmetrization and, more generally, polarization of a semigroup Koenigs domain slow the semigroup’s convergence to its Denjoy–Wolff point, quantified via harmonic-measure bounds. By rotating to normalize the Denjoy–Wolff points and applying diameter estimates together with harmonic-measure comparisons, the authors obtain explicit rate inequalities: $|φ_t(0)-τ| ≤ 4π\,|φ_t^lat(0)-τ^lat|$ for Steiner symmetrization and $|φ_t(0)-τ| ≤ 2π\,|\widehat{φ}_t(0)-\widehat{τ}|$ for polarization. The results hinge on the conformal invariance and monotonicity of harmonic measure, together with Baernstein-type symmetrization and Solynin-type polarization inequalities, linking domain geometry to dynamical rates. These findings provide a quantitative bridge between geometric transformations of Koenigs domains and the speed of semicircular dynamics toward the Denjoy–Wolff point, with potential applications to domain-geometry-based rate estimates in geometric function theory.

Abstract

Let $(φ_t)$, $t\ge 0$, be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$. Let $Ω$ be its Koenigs domain and $τ\in \partial \mathbb{D}$ be its Denjoy-Wolff point. Suppose that $0\in Ω$ and let $Ω^\sharp$ be the Steiner symmetrization of $Ω$ with respect to the real axis. Consider the semigroup $(φ_t^\sharp)$ with Koenigs domain $Ω^\sharp$ and let $τ^\sharp$ be its Denjoy-Wolff point. We show that, up to a multiplicative constant, the rate of convergence of $(φ_t^\sharp)$ is slower than that of $(φ_t)$; that is, for every $t>0$, $|φ_t(0)-τ|\leq 4π\, |φ_t^\sharp(0)-τ^\sharp|$. The main tool for the proof is the harmonic measure.

Figures (4)

• Figure 1: A domain $\Omega$ convex in the positive direction and its symmetrization $\Omega^\sharp$.
• Figure 2: A domain $\Omega$ convex in the positive direction and its polarization $\widehat{\Omega}$.
• Figure 3: The domains $\Omega^\sharp_t$ and $\widetilde{\Omega_t}$.
• Figure 4: The hyperbolic geodesic $\widehat{\Gamma_t}$ and the domain $\mathop{\mathrm{\mathbb{D}}}\nolimits_t$.

Theorems & Definitions (11)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 4.1
• Lemma 4.2