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arxiv: 2605.05184 · v1 · submitted 2026-05-06 · 🧮 math.DS

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Boundaries of Baker domains of entire functions. A finer approach

Anna Jov\'e, {\L}ukasz Pawelec

Pith reviewed 2026-05-08 16:05 UTC · model grok-4.3

classification 🧮 math.DS
keywords Baker domainsentire functionsinner functionsboundary dynamicsmeasure theoryDenjoy-Wolff pointtranscendental dynamicsparabolic points
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The pith

For transcendental entire functions with doubly parabolic Baker domains, the boundary dynamics admit a precise measure-theoretic description when the Denjoy-Wolff point is not a singularity.

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

The paper focuses on transcendental entire functions possessing doubly parabolic Baker domains under the condition that the Denjoy-Wolff point of the associated inner function is not a singularity. It establishes a detailed measure-theoretic account of the dynamics on the boundaries of these domains. This description arises from new strengthened results about the radial extensions of one-component doubly parabolic inner functions. A sympathetic reader would care because the work yields sharper information about both the topology and the dynamical behavior on those boundaries, with explicit improvements to prior models for the Baker domain of the function z plus e to the minus z.

Core claim

For transcendental entire functions having doubly parabolic Baker domains such that the Denjoy-Wolff point of the associated inner function is not a singularity, the boundary dynamics can be described in a very precise measure-theoretic way by applying strengthened results on the radial extension of one-component doubly parabolic inner functions.

What carries the argument

The radial extension of one-component doubly parabolic inner functions, which carries the argument by supplying the measure-theoretic control on boundary orbits and strengthening earlier thermodynamic-formalism results.

Load-bearing premise

The Denjoy-Wolff point of the associated inner function is not a singularity.

What would settle it

Identify a concrete transcendental entire function with a doubly parabolic Baker domain whose associated inner function has a singular Denjoy-Wolff point and check whether the boundary dynamics still obey the stated measure-theoretic description.

Figures

Figures reproduced from arXiv: 2605.05184 by Anna Jov\'e, {\L}ukasz Pawelec.

Figure 1.1
Figure 1.1. Figure 1.1: Schematic representation of targets. Indeed, for the Baker domain U of f(z) = z + e −z , the figure shows the image of two arcs in the unit circle, each of them bounded by two rays landing at infinity in the dynamical plane. Hence, given such an invariant Fatou component, it is a natural to examine the distribution of orbits on ∂U with respect to a given target. A first step on this direction are the res… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: For the same Baker domain as above, a finite target (in purple) and an infinite target (in green). Note that 1 is the Denjoy-Wolff point of the associated inner function. Inspired by results of Thaler and Zweim¨uller [TZ06] on distributional limit theorems in infinite ergodic theory, we prove the following finer results for Baker domains of finite degree (in fact, the assumption of finite degree can be w… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: On the left, Boole’s map (where a change of coordinates is applied to bring it to a finite interval). Such a map has two branches, both containing a parabolic point (at 0 and at 1). On the right, a AFN-system composed by five branches, two of them with a parabolic point (at 0 and at 3) –in red–, and the others hyperbolic –in blue–. and Zα – if necessary those can be found in the cited paper. However, und… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: On the left, the graph of x 7→ 1/ tan(1/x), which is the radial extension of a doubly parabolic inner function; 0 is a singularity of the map and ∞ is the doubly parabolic fixed point. On the right, the function x 7→ tan x (which is conjugate to the previous function, but now the Denjoy-Wolff point is at 0 and the singularity is at infinity). The easiest example of a doubly parabolic inner function is th… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The graph of Boole’s map T(x) = x − 1/x. 3.3. Mapping structure around the Denjoy-Wolff point. Most of our arguments rely on the linearization around the Denjoy-Wolff point, following the ideas of [IU23]. More precisely, if h is a doubly parabolic inner function with Denjoy-Wolff point at ∞, this means that the normal form of the associated inner function around ∞ point is the one corresponding to a para… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The points · · · < p− 3 < p− 2 < p− 1 and p + 1 < p+ 2 < p+ 3 < . . . for Boole’s map T(x) = x − 1/x. Note that p − 1 and p + 1 coincide (and equal 0), and z ± = p ± 2 . 3.4. Finite degree doubly parabolic Blaschke products. Inner functions of finite degree are finite Blaschke products, i.e. they have the form g : D → D, g(z) = e iθ Yn k=1 ak − z 1 − akz , where θ ∈ [0, 2π), ak ∈ D, and n ∈ N is the degr… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Graph of the function T : R → R, x 7→ x − 1 x−1 − 1 x+1 − 1 x−2 − 1 x+2 , which is the radial extension of a doubly parabolic Blaschke product of finite degree (seen in H). Its hyperbolic branches are drawn in red, while the parabolic ones in blue. Note that a1 = p − 1 = −2 and an = p + 1 = 2. Recall that D is the set (z −, z+), where z − is the preimage of zero placed most to the left, and z + is the on… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The points · · · < p− 3 < p− 2 < p− 1 and p + 1 < p+ 2 < p+ 3 < . . . for Boole’s map T(x) = x − 1/x view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Dynamical plane for f(z) = z + e −z . In red, the Julia set of f. In beige, the Baker domain contained in the strip {−π < Im z < π}. In black, the rest of the Fatou set of f. The only critical point on the strip (0) is also marked, as well as the corresponding critical value (1). • We define a measure ν in the space Σ2 as the push-forward of the measure ωU on ∂U. Note that such a measure is finite, but i… view at source ↗
read the original abstract

We consider transcendental entire functions having doubly parabolic Baker domains, such that the Denjoy-Wolff point of the associated inner function is not a singularity. We describe in a very precise way the dynamics on the boundary from a measure-theoretical point of view. Applications of such results lead to a better understanding of the topology and the dynamics on the boundaries. In particular, we improve some of the results in [N. Fagella and A. Jov\'e, A model for boundary dynamics of Baker domains], for the Baker domain of $z+e^{-z}$. In fact, our conclusions are obtained by applying new results established here on the dynamics of the radial extension of one component doubly parabolic inner functions, which strengthen those of [O. Ivrii and M. Urba\'nski, Inner functions, composition operators, symbolic dynamics and thermodynamic formalism].

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 / 2 minor

Summary. The manuscript considers transcendental entire functions with doubly parabolic Baker domains where the Denjoy-Wolff point of the associated inner function is not a singularity. It establishes new results on the dynamics of radial extensions of one-component doubly parabolic inner functions, strengthening those of Ivrii and Urbański. These results are applied to give a precise measure-theoretic description of boundary dynamics, improving the understanding of topology and dynamics on such boundaries, with a concrete application that refines some results of Fagella and Jovè for the Baker domain of z + e^{-z}.

Significance. If the new results on radial extensions hold, the work provides a finer measure-theoretic framework for boundary dynamics of Baker domains, building directly on cited prior literature with a concrete improvement for the standard example z + e^{-z}. This strengthens the link between inner-function dynamics and entire-function iteration without introducing free parameters or ad-hoc constructions, and the explicit application adds practical value for further studies in complex dynamics.

minor comments (2)
  1. The abstract refers to 'one component doubly parabolic inner functions' without a brief inline clarification or forward reference to the definition; this could be added for accessibility.
  2. The specific improvements over the results in [N. Fagella and A. Jovè] should be enumerated explicitly (e.g., which theorems or statements are strengthened) in the introduction or a dedicated comparison subsection.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation in application; central derivation independent

full rationale

The paper establishes new results on the radial extension dynamics of one-component doubly parabolic inner functions, explicitly strengthening the external Ivrii-Urbański framework. These new results are then applied to improve an earlier model from the authors' own prior work with Fagella. The self-citation is confined to the example application for z+e^{-z} and is not load-bearing for the measure-theoretic boundary description or the strengthened inner-function theorems. No self-definitional reductions, fitted inputs renamed as predictions, or ansatz smuggling appear in the stated chain. The non-singularity assumption on the Denjoy-Wolff point is stated explicitly and remains external.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard background from complex dynamics and inner function theory. No free parameters, invented entities, or ad hoc axioms are apparent from the abstract description.

axioms (2)
  • domain assumption Standard properties of doubly parabolic inner functions and their radial extensions hold as in prior literature.
    Invoked to establish the new dynamical results on boundaries.
  • domain assumption The Denjoy-Wolff point of the inner function is not a singularity.
    Explicitly stated as the setting for the functions considered.

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Reference graph

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