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arxiv: 1907.07098 · v1 · pith:S3YTAV7Unew · submitted 2019-07-16 · 🧮 math.CV · math.DS

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

Filippo Bracci This is my paper

Pith reviewed 2026-05-24 20:29 UTC · model grok-4.3

classification 🧮 math.CV math.DS
keywords non-elliptic semigroupsholomorphic self-mapsunit discorbit convergencetotal speedorthogonal speedtangential speed
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The pith

Three quantities measure convergence speeds of orbits under non-elliptic holomorphic semigroups on the unit disc.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the total speed, orthogonal speed, and tangential speed for orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disc. It establishes explicit relations among these three quantities. From the relations, further properties of the orbits follow directly. A reader would care because the decomposition separates radial and tangential components of approach to the boundary, giving a sharper description of asymptotic behavior than a single rate.

Core claim

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

What carries the argument

The total speed, orthogonal speed, and tangential speed of an orbit, which decompose its rate of convergence to the boundary.

If this is right

  • The relations among the three speeds constrain the possible asymptotic behaviors of the orbits.
  • Finiteness or infiniteness of one speed implies corresponding statements about the others.
  • Properties of orbit convergence can be read off directly from the values of the speeds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition could be attempted for discrete iterates rather than continuous semigroups.
  • The speeds might distinguish different boundary fixed-point behaviors without computing the full semigroup.
  • Numerical approximation of the three speeds on sample maps would test whether the relations survive discretization.

Load-bearing premise

The three speed quantities are well-defined and finite for the orbits under consideration.

What would settle it

An explicit orbit of a non-elliptic semigroup for which the claimed relations between total speed, orthogonal speed, and tangential speed fail to hold.

read the original abstract

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disc—the total speed, the orthogonal speed, and the tangential speed—and establishes relations among them along with inferences that follow from those relations.

Significance. If the definitions prove well-defined and finite and the stated relations hold, the work supplies a new set of tools for quantifying different aspects of convergence speed in holomorphic semigroup dynamics. Distinguishing total, orthogonal, and tangential components allows finer analysis of orbit behavior than is available from existing notions of hyperbolic distance or angular derivative alone.

minor comments (3)
  1. The abstract is terse; a sentence indicating the principal theorems or the form of the relations proved would help readers locate the contribution quickly.
  2. Notation for the three speeds should be introduced with a single consistent symbol set (e.g., v_tot, v_orth, v_tan) and used uniformly from the first definition onward.
  3. A brief comparison table or diagram relating the new speeds to classical quantities such as the hyperbolic distance along the orbit would clarify the novelty for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces three new quantities (total speed, orthogonal speed, tangential speed) as definitions for orbits of non-elliptic semigroups and proves relations and inferences among them. No load-bearing step reduces a result to a fitted parameter, self-citation chain, or input by construction; the well-definedness and finiteness of the quantities are part of the theorems to be proved rather than presupposed. The work is self-contained against external benchmarks with no renaming of known results or smuggled ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the definitions themselves constitute the contribution.

pith-pipeline@v0.9.0 · 5556 in / 972 out tokens · 16698 ms · 2026-05-24T20:29:41.083528+00:00 · methodology

discussion (0)

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Reference graph

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