Speeds of convergence of orbits of non-elliptic semigroups of holomorphic self-maps of the unit disk

Filippo Bracci

doi:10.17951/a.2019.73.2.21-43

Abstract

We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disk, the total speed, the orthogonal speed, and the tangential speed and show how they are related and what can be inferred from those.

Highlights

Continuous semigroups of holomorphic self-maps of the unit disk D, or for short, semigroups in D, have been studied since the beginning of the previous century and are still a subject of interest, from the dynamical point of view, the analytic point of view, and the geometric point of view, and for different applications.In this paper, we consider non-elliptic semigroups in D
The total speed can be decomposed, up to a universal additive constant, as the sum of two other quantities, the orthogonal speed vo(t) and the tangential speed vT (t). This is a general fact of hyperbolic geometry which we prove in Section 3: given a curve γ : [0, +∞) → D starting from 0, converging to point σ ∈ ∂D, the orthogonal projection of γ(t) over (−1, 1)σ is the point π(γ(t)) ∈ (−1, 1)σ such that ω(π(γ(t)), γ(t)) = inf{ω(rσ, γ(t)) : r ∈ (−1, 1)}
Besides settling the notions of speeds and proving the aforementioned results, in this paper we provide a direct computation of total, orthogonal and tangential speeds in some cases

Summary

Introduction

Continuous semigroups of holomorphic self-maps of the unit disk D, or for short, semigroups in D, have been studied since the beginning of the previous century and are still a subject of interest, from the dynamical point of view, the analytic point of view, and the geometric point of view, and for different applications. The total speed can be decomposed, up to a universal additive constant, as the sum of two other quantities, the orthogonal speed vo(t) and the tangential speed vT (t) This is a general fact of hyperbolic geometry which we prove in Section 3: given a curve γ : [0, +∞) → D starting from 0, converging to point σ ∈ ∂D, the orthogonal projection of γ(t) over (−1, 1)σ is the (unique) point π(γ(t)) ∈ (−1, 1)σ such that ω(π(γ(t)), γ(t)) = inf{ω(rσ, γ(t)) : r ∈ (−1, 1)}. In case of a non-elliptic semigroup (φt), we define the orthogonal speed vo(t) := ω(0, π(φt(0))), where π is the orthogonal projection on (−1, 1)τ , where τ is the Denjoy–Wolff point of (φt). The paper ends with a section of open questions which naturally arise from the developed theory

Hyperbolic geometry in simply connected domain

Continuous non-elliptic semigroups of holomorphic self-maps of the unit disk

Speeds of non-elliptic semigroups

Total speed of convergence

Orthogonal speed of convergence of parabolic semigroups

Open Questions The previous results give rise to the following questions