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arxiv: 2606.20147 · v1 · pith:VQ24EKJN · submitted 2026-06-18 · math.DS · math.CV

Inner functions associated to lifts of transcendental entire functions

Eleni Betsakou This is my paper

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classification math.DS math.CV
keywords inner functionstranscendental entire functionsFatou componentsliftswandering domainsRiemann mapscomplex dynamics
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The pith

If f is a lift of a transcendental entire function h, then the inner function associated to f on U is obtained from the inner function associated to h on the lifted component G.

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

The paper establishes a method to compute associated inner functions for transcendental entire functions that arise as lifts. When f lifts h, with U the component lifting from G, the inner function g_f on U relates to g_h on G through the covering relation and Riemann maps. This extends an earlier theorem to both finite- and infinite-degree cases and covers forward-invariant components as well as wandering domains. Readers would care because explicit inner functions remain scarce for infinite-degree maps, and the lift construction supplies them for many previously studied examples.

Core claim

If f is a lift of a transcendental entire function h, then an inner function associated to f restricted to U can be obtained by relating it to an inner function associated to h restricted to G, where G is the Fatou component that lifts to U. The relation holds whether the components are forward-invariant or wandering, and whether the degree on the component is finite or infinite.

What carries the argument

The lift relation between f and h together with the Riemann maps from the disk to U and G, which compose with the covering map to transfer the inner function from the base to the lift.

If this is right

  • The construction applies directly to several transcendental entire functions already studied in the literature.
  • The same relation works for both finite-degree and infinite-degree restrictions to the component.
  • The result covers forward-invariant Fatou components and wandering domains alike.
  • It recovers and extends the main statement of the theorem by Evdoridou, Rempe and Sixmith.

Where Pith is reading between the lines

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

  • The method may yield explicit inner functions for additional families of entire functions that admit lifts but lack closed-form expressions today.
  • If similar lift relations exist for meromorphic functions, the same transfer could apply beyond the entire case.
  • The reduction might simplify iteration studies in the infinite-degree setting by moving computations to the base function h.

Load-bearing premise

The Fatou component U lifts from G under the given lift relation, and the Riemann maps compose appropriately with f, h, and the covering map.

What would settle it

A concrete lift f of h together with explicit Riemann maps where the resulting g_f fails to equal the composition that relates it to g_h.

Figures

Figures reproduced from arXiv: 2606.20147 by Eleni Betsakou.

Figure 1
Figure 1. Figure 1: Constructing an associated inner function to f|U use the same Riemann map, and then gf is called a dynamically associated inner function to f|U and it is unique up to a conformal conjugacy. There is a growing interest in inner functions associated to Fatou components. One reason for this is that it is often simpler to study a problem in the unit disc, using inner functions, and then transfer the results to… view at source ↗
Figure 2
Figure 2. Figure 2: The Fatou (grey) and Julia (black) sets of the function f(z) = 1 + z + e −z . the aforementioned examples, the Fatou functions exhibit additional complexity; the set of singular values - the closure of the set of critical and finite asymptotic values - of fλ is not compactly contained in the Baker domain. This fact makes it difficult to calculate a dynamically associated inner function, since there is no g… view at source ↗
Figure 3
Figure 3. Figure 3: Proving that U is connected For the converse statement, suppose that G is simply connected and U := π −1 a (G) is connected. The definition of U implies that it contains all the preimages under πa of any point in G (except for 0, when 0 ∈ G). Since there are infinitely many such preimages, we deduce that πa is not injective in U. Now, if 0 ∈/ G, then the restriction πa|U : U → G is a covering map, because … view at source ↗
Figure 4
Figure 4. Figure 4: Showing that f has infinite degree in U (m = 3 here) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Showing that h has infinite degree in G preimages of zk under f in U, for each k ∈ Z. Fix k ∈ Z. If there exist n, n′ ∈ N such that (2.5) zk, n′ = zk, n + 2lπi a , for some l ∈ Z \ {0}, then (2.3) implies that m = 0 and f|U is 2πi a -periodic. Thus, for each k ∈ Z, the set of preimages of zk under f in U, {zk, n}n∈N, contains a subset {v j k }j∈Z, such that v j+1 k = v j k + 2πi a , for all j ∈ Z, and so π… view at source ↗
Figure 6
Figure 6. Figure 6: Constructing the dynamically associated inner function gf to f|U Let z ∈ H. Then there exists a point w ∈ H such that z = M−1 (w). Thus, we have that gh(ˆp(z)) = gh(ˆp(M−1 (w))) = gh(˜p(w)) = ˜p(˜gf (w)) = ˜p(˜gf (M(z))) = ˜p(M(gf (z))) = ˆp(gf (z)), so gh(e 2iz) = e 2igf (z) ⇔ e iθe 2miz · exp  − X q j=1  cj · e iθj + e 2iz e iθj − e 2iz  = e 2igf (z) ⇔ 2igf (z) = iθ + 2miz − X q j=1  cj · e iθj + e … view at source ↗
Figure 7
Figure 7. Figure 7: The dynamical planes of the function h(w) = w 2 e −w and of its lift f(z) = 2z + e −z . proof of [KU05, Lemma 2.7], we can show that there exists δ ∈ (0, π 3 ] such that J(f) ⊂ [ k∈Z  {z ∈ C : Rez ≤ 0.6, 2kπ + δ < Imz < (2k + 1)π} ∪ {z ∈ C : Rez ≤ 0.6, (2k + 1)π < Imz < 2(k + 1)π − δ}  , and J(f) consists of disjoint curves tending to ∞, each homeomorphic to [0, +∞), by [RRS10, Theorem 1.3] (see Figure 7… view at source ↗
Figure 9
Figure 9. Figure 9: The action of φ on H Having established the above results, we can choose α appropriately, so that κ = π, and hence we obtain (6.2). We are now ready to calculate σ and θ. It follows from (6.2) that φ(lk) = η−k, with (6.10) φ(kπ + iα) = −2kπi, φ(l + k, α) = η + −k and φ(l − k, α) = η − −k , for all k ∈ Z (see [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
read the original abstract

Let $f$ be a transcendental entire function, $V$ be a simply connected Fatou component of $f,$ and $U$ be a Fatou component with $f(U)\subset V.$ There is a natural way to associate $f|_U$ to an inner function, namely a function $g_f:=\psi^{-1}\circ f\circ\varphi,$ where $\varphi:\mathbb{D}\to U$ and $\psi:\mathbb{D}\to V$ are Riemann maps. Inner functions have been used as a tool in the study of the iterates of transcendental entire, and more recently meromorphic, functions. However, there are only a few examples where associated inner functions have been calculated explicitly, with the case where $f$ has infinite degree in $U$ being the least well understood and more complicated. In this paper, we introduce a general method for calculating associated inner functions to a wide class of entire functions arising as `lifts'. In particular, if $f$ is a lift of a transcendental entire function $h,$ we show that an inner function associated to $f|_U$ can be obtained by relating it to an inner function associated to $h|_G,$ where $G$ is the Fatou component that lifts to $U.$ This result significantly generalises the main part of a theorem by Evdoridou, Rempe and Sixmith, and can be applied to several functions that have been studied so far. In both finite- and infinite-degree settings, the results hold for forward-invariant Fatou components as well as for wandering domains.

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 paper claims that if f is a transcendental entire function that is a lift of another such function h, then an inner function g_f associated to f restricted to a Fatou component U (via Riemann maps) can be obtained by relating it to the inner function associated to h restricted to the lifted component G. The result is stated to hold in both finite- and infinite-degree settings and for both forward-invariant and wandering domains, generalizing the main theorem of Evdoridou–Rempe–Sixmith.

Significance. If the derivation holds, the work supplies a systematic method for computing associated inner functions for an entire class of transcendental entire maps arising as lifts. This extends the limited set of explicit examples currently available, particularly in the infinite-degree case, and directly applies to several previously studied functions. The uniform treatment of invariant and wandering components is a notable strengthening.

minor comments (2)
  1. [§1] §1 (Introduction): the sentence claiming the result 'can be applied to several functions that have been studied so far' would benefit from an explicit list or forward reference to the examples treated in §4 or §5.
  2. Notation: the covering map relating U to G is introduced without an explicit symbol; adding a consistent notation (e.g., π: U → G) would improve readability when the Riemann-map composition is written out.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the work provides a systematic method for computing associated inner functions in both finite- and infinite-degree settings and for both invariant and wandering domains. We note the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via lift relation

full rationale

The paper's central claim constructs an inner function for the lift f|U by composing Riemann maps with the given lift relation to the base function h|G. This step is defined directly from the covering/lift property between domains U and G together with the standard definition of associated inner functions g_f = ψ^{-1} ∘ f ∘ ϕ; it does not reduce to a fitted parameter, self-definition, or load-bearing self-citation. The result is presented as a generalization of an external theorem (Evdoridou–Rempe–Sixmith) whose authors do not overlap with the present author, and the abstract states the construction applies uniformly to finite/infinite degree and invariant/wandering cases without invoking any uniqueness theorem or ansatz imported from the same authors. No equation or normalization in the provided material equates the output inner function to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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