Entire functions with prescribed singular values

Luka Boc Thaler

REVIEW 2 major objections 5 minor 12 references

Entire functions with prescribed singular values

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that every closed set containing 0 and one more point is the singular-value set of a locally univalent entire function in a new explicit class, and that this class is dense in the non-vanishing entire functions.

desk verdict A new construction class for entire functions with prescribed singular values; the central theorem is sound modulo a repairable overstatement in Section 3 and a minor gap in Example 3. read the letter →

arxiv 1908.06026 v3 pith:DMVMREZ4 submitted 2019-08-16 math.CV

Luka Boc Thaler This is my paper · ORCID

classification math.CV MSC 30D2030D0537F10

keywords entirefunctionssingularvaluesasymptoticlocallyunivalentclassEiteratedexponentialsEremenko-LyubichFatouset

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a class $\mathcal{E}$ of entire functions (complex-differentiable maps defined on all of $\mathbb{C}$) generated by infinite backward compositions of rescaled exponentials, meaning $F_0\in\mathcal{E}$ when $F_n(z)=\lambda_{n+1}e^{F_{n+1}(z)}$ for a sequence of nonzero constants $\lambda_n$. Its main theorem states that every closed set $U\subset\mathbb{C}$ containing $0$ and at least one other point is the singular-value set $S(f)$ of a locally univalent entire function $f\in\mathcal{E}$, and that $f$ can be chosen uniformly close to the exponential map on any compact set. Because adding a constant translates the singular-value set, the theorem recovers the classical result that every closed subset of the plane occurs as the singular-value set of a locally univalent entire function. The point is methodological: the functions are explicit uniform limits of approximants with known singular values, so the construction tracks a quantity that is normally unstable under uniform convergence.

What carries the argument

The mechanism has two parts. First, the class $\mathcal{E}$ is generated by the identities $F_n(z)=\lambda_{n+1}e^{F_{n+1}(z)}$, giving the telescoping relation $F_0=E_{(0,n)}\circ F_n$; this makes every orbit point $E_{(0,k)}(0)$ an asymptotic value of $F_0$. Second, the explicit approximants $f_{k,n}$ and their inverse branches $g_{I_n}(z)=(L_{I_n}(z)-b_n)b_0\cdots b_{n-1}$, where $L_{I_n}$ is a composition of logarithmic branches $L^\lambda_k(z)=\log z-\log\lambda+2k\pi i$, control which asymptotic values survive the uniform limit. The load-bearing tool is Lemma 7: if $g_{I_n}$ converges uniformly on a neighbourhood of a point $\zeta$ outside the orbit, then the index tuple eventually has the form $(k_1,\ldots,k_j,m_{j+1},\ldots,m_n)$. This forces the limiting inverse branches to be exactly the ones anchored by the first $j$ choices, so a point outside the dense orbit cannot be an asymptotic value of $F_0$.

What would settle it

Take $U=\{0,1\}$ and choose the constants $\lambda_n$ so that the iterated points $E_{(0,n)}(0)$ accumulate at $1$; then inspect the inverse branches $g_{I_n}$ of the explicit approximants $f_{0,n}$ near $1$. If no sequence of these branches converges uniformly on a fixed neighbourhood to a branch of the limiting $F_0$, then $1$ is not an asymptotic value and $S(F_0)$ misses $1$, contradicting Theorem 1 for this choice of $U$.

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Extended reading notes

Core claim

The central claim is Theorem 1: given any closed $U\subset\mathbb{C}$ containing $0$ and one further point, any compact set $K$, and any $\varepsilon>0$, there is a locally univalent $f\in\mathcal{E}$ with $S(f)=U$ and $\|f-E_1\|_K<\varepsilon$, where $E_1(z)=e^z$. The construction first chooses constants $\lambda_n$ so that the orbit points $E_{(0,n)}(0)$, with $E_{(0,n)}=E_{\lambda_1}\circ\cdots\circ E_{\lambda_n}$ and $E_\lambda(z)=\lambda e^z$, are dense in $U$. The approximants are $f_{0,n}(z)=E_{(0,n)}((b_n+z)/(b_0\cdots b_{n-1}))$, where $b_0=1$ and $b_n=2\pi i m_n+\log b_{n-1}-\log\lambda_n$ for rapidly increasing integers $m_n$. They converge uniformly on compact sets to a nonconstant locally univalent $F_0\in\mathcal{E}$, and each orbit point $E_{(0,k)}(0)$ is an asymptotic value of $F_0$. The proof that no other points are asymptotic values uses inverse branches of $f_{0,n}$: a stabilization lemma forces every convergent sequence of inverse branches to agree eventually with a single branch of $F_0$, so no accidental asymptotic values arise. Hence $S(F_0)$ equals the closure of the dense orbit, namely $U$.

Load-bearing premise

The proof relies on every point excluded from the singular set having a fixed neighbourhood on which all inverse functions of the approximating maps are defined with uniform bounds, including at points where the chosen iterated points accumulate.

Editorial extensions

If this is right

Every closed set $V\subset\mathbb{C}$ is the singular-value set of some locally univalent entire function: the one-point cases are covered by $e^z+c$, the empty case by the identity map, and all other cases by Theorem 1 followed by a constant shift.
For any closed $U$ containing $0$, the exponential functions $E_\lambda(z)=\lambda e^z$ lie in the closure of the class $\{f:S(f)=U\}$, so prescribed singular-value sets can be realized while staying uniformly close to exponentials.
The class $\mathcal{E}$ is dense in the space of non-vanishing entire functions, with the zero function also in the closure, giving many explicit uniform limits with controlled singular values.
Functions in $\mathcal{E}$ can have empty Fatou set with finite singular values $\{0,1\}$, so the construction produces examples with completely repelling postsingular dynamics, and it also produces functions with nonempty Fatou set.
The class $\mathcal{E}$ is closed under composition and under composition with arbitrary entire functions, so it forms a compositional reservoir of explicit maps whose singular values remain controlled, intersecting both the Speiser class and the Eremenko-Lyubich class.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit approximants suggest a testable extension: prescribe, along with $S(f)=U$, finitely many local data such as $f(0)$ and $f'(0)$; the paper already constructs functions in $\mathcal{E}$ with $f(1)=1$ and $f'(1)=\lambda$, so a systematic interpolation theorem is plausible.
The branch-stabilization mechanism is not obviously tied to exponentials; replacing $E_\lambda$ by another one-parameter family of covering maps could yield analogous classes with controlled singular values if the constants can force inverse branches to stabilize.
One could study quantitative versions of the theorem, such as how the rate of approximation to $e^z$ on growing compact sets depends on the rapidly increasing integers $m_n$ and on the geometry of $U$; the paper leaves this trade-off unexplored.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main thing you should know: this paper is real, not a curiosity. It introduces a class E of entire functions (those expressible as F0 with F_n = λ_{n+1}e^{F_{n+1}}), shows E is dense in the non-vanishing entire functions, and proves that every closed set containing 0 and at least one other point is the singular-value set of some locally univalent function in E. That implies Heins's theorem as a corollary, but with a genuinely different proof mechanism: explicit towers of exponentials whose inverse branches can be tracked.

The best part is the proof of Theorem 1. The author constructs F0 as a uniform limit of f_{0,n}, where each f_{0,n} is in the Speiser class with singular values exactly the first n points of a chosen orbit. The inverse-branch argument in Lemma 7 is clever: it forces the winding indices to stabilize, and the estimates (15)–(16) show the inverse branches converge. The claim S(F0) = V follows cleanly once you have convergence of all inverse branches outside V. This is a real advance over Heins's uniformization construction, and the recovery of Heins is not a trivial consequence.

The soft spot is in Section 3, as you flagged. The paper defines Ω as the complement of the orbit and asserts every ζ ∈ Ω has a fixed neighbourhood U_ζ on which all inverse branches of every f_{0,n} are defined for all n. That is false when ζ is an accumulation point of the orbit: any neighbourhood of ζ contains orbit points that are singular values of f_{0,n} for all large n. The fix is easy: replace Ω with the complement of V (the orbit closure). For ζ outside V, positive distance to the orbit gives the needed neighbourhood, and that is all the proof needs to get S(F0) ⊆ V. So the theorem survives, but the written proof contains an overstatement that must be corrected.

The second issue is minor: Example 3 claims the Fatou set is empty from finite postsingular set plus |f'(1)|>1. That doesn't follow without an argument (or a citation), and the text just asserts it. It does not affect the main theorem, but a referee should ask for a justification.

Net: this deserves a serious referee. The construction is new, the main theorem is correct in substance, and the gaps are repairable. I would send it to review and ask for a revision that fixes the neighbourhood statement and fleshes out Example 3.

Referee Report

2 major / 5 minor

Summary. The paper introduces a class E of entire functions F0 for which there exist entire Fn and nonzero constants lambda_n satisfying Fn = lambda_{n+1} exp(F_{n+1}) for all n. The main result (Theorem 1) states that for every closed set U containing 0 and at least one further point, and for every compact K and epsilon>0, there is a locally univalent f in E with S(f)=U and ||f-E_1||_K<epsilon. The proof first constructs, for any prescribed nonzero lambda_n, a uniform limit F0 of explicit compositions of exponentials, and then chooses the lambda_n so that the points E_{(0,n)}(0) form a dense subset of U. A separate inverse-branch argument is used to show that no other singular values appear. From Theorem 1 the author derives Corollary 2 (Heins' theorem that every closed set is the set of singular values of a locally univalent entire function), further closure properties of E, and two examples, one asserting the existence of functions in E with nonempty Fatou set and one asserting the existence of functions with empty Fatou set.

Significance. If the proof is repaired, this is a valuable and original contribution. The construction is explicit and self-contained, avoiding uniformization and quasiconformal folding; the functions are obtained as uniform limits of concrete approximants, and the paper shows that despite the known instability of the singular-value set under limits, the particular limiting process can be used to prescribe S(f) exactly. This gives a new proof of a classical theorem of Heins. The paper also correctly identifies useful structural properties of the class E, such as closure under composition and dense embedding into the space of non-vanishing entire functions, and the proof is based on standard tools (Picard, Hurwitz, Rouche) with no fitted parameters. The advertised dynamical consequences (Corollaries 8 and 9 and the Fatou-set examples) broaden the interest of the paper, provided Example 3 is fully justified.

major comments (2)

[Section 3, paragraph after defining Omega] The assertion after the definition of Omega, that every zeta in Omega has a neighbourhood U_zeta on which all inverse branches of f_{0,n} are well defined for all n>0, is false. If zeta is an accumulation point of the orbit {E_{(0,k)}(0)} that is not itself an orbit point, then zeta lies in Omega, but every neighbourhood of zeta contains some E_{(0,ell)}(0), and for all n>ell this point is a singular value of f_{0,n}; hence no inverse branch of f_{0,n} can be defined on the whole neighbourhood. This false claim is load-bearing as written, because Lemma 7 and the estimates leading to (15) and (16) are applied to all of Omega, and the proof concludes S(F0) subset of the orbit closure. The gap is repairable: replace Omega by C\V. Since V is closed, every zeta outside V has positive distance to V and therefore to every orbit point, so a fixed U_zeta avoiding all excluded finite sets exists; the remaining argument then proves S(F0) subset of V, which is exactly what is needed. The repair is local, but as it stands the proof contains an unsupported intermediate claim.
[Section 4, Example 3] The statement 'hence the Fatou set of f is empty' is not justified. From |f'(1)|>1 one only obtains that 1 is a repelling fixed point. To conclude that F(f)=emptyset one must rule out attracting, parabolic, and Siegel periodic Fatou components as well as wandering and Baker domains. The finiteness of S(f) and the postsingular set {0,1} make this plausible and probably standard, but the argument is not written and no citation is given. Please either supply a short proof or cite a theorem that, for an entire function with finite postsingular set all of whose periodic points are repelling, the Fatou set is empty.

minor comments (5)

[Abstract] The phrase 'and its is dense' should read 'and it is dense'.
[Section 2, proof of Theorem 3] The sentence 'Let (r_n) be an increasing sequence of integers' should say 'real numbers' or 'positive numbers', since the radii of the disks need not be integers.
[Section 2, alternative proof of Proposition 6] The word 'Pioncaré' is a typo for 'Poincaré'.
[Section 4, Example 3] Please spell out the choice of epsilon in the Rouché argument and the Cauchy estimate for the derivative; the sentence 'using Cauchy estimates we obtain |f'(1)|>1' is too terse for the claimed quantitative conclusion.
[References] Reference [5] contains the garbled phrase 'Konvergenzwert st'; the intended German term should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction identifies S(F0) with the closure of a pre-chosen orbit by independent inverse-branch arguments rather than by assuming the target result.

full rationale

The central construction is self-contained. Theorem 3 builds F0 and the whole sequence (F_n) from an arbitrary nonzero sequence (λ_n) via explicit approximants f_{k,n}, with convergence estimates that do not refer to the prescribed set V. In Section 3, V enters only through the choice of a dense sequence a_n and the inductive choice of λ_n satisfying E_{(0,n)}(0)=a_n; this is a standard surjectivity argument for functions of the form C exp(B e^w), not a fitted parameter renamed as a prediction. The inclusion V ⊆ S(F0) is derived from the composition identity (1) applied to F0 = E_{(0,n)}∘F_n, and the reverse inclusion is then verified by showing that inverse branches of F0 on Ω are uniform limits of inverse branches of the approximants f_{0,n}, using Lemma 7 and the estimates (15)–(16). This is an independent proof step, not a restatement of the conclusion. No load-bearing self-citation appears: the cited Poincaré-function material is standard and auxiliary, and the Heins theorem is presented as a corollary rather than used as an input. The supplied text does not exhibit any equation or construction that reduces to its own target. A possible correctness gap in Section 3 concerning the existence of a common inverse-branch neighbourhood on all of Ω is a validity concern, not a circularity, and thus does not affect the circularity score.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central construction rests on standard complex analysis theorems and explicit choices of sequences. No new physical or mathematical entities are postulated. The proof uses auxiliary sequences m_n and lambda_n, but these are construction parameters, not empirical fits.

free parameters (3)

integer sequence (m_n)
Chosen sufficiently fast to make f_{k,n} converge and to force the inverse-branch bounds in equations (15) and (16). Existence of such a sequence is enough; no unique value is fitted to data.
sequence (lambda_n)
Chosen recursively so that E_{(0,k)}(0) is dense in the prescribed closed set V. These are construction parameters, not fitted constants.
dense sequence (a_n) in V
A countable dense subset of V with a_0 = 0 and a_n nonzero for n at least 1. The choice is auxiliary to the construction.

assumptions (4)

standard math Picard's theorem: a nonconstant entire function omitting two values is constant; hence a nonvanishing nonconstant entire function is surjective onto C-star.
Used in Section 3 to solve E_{(0,k)}(0) = a_k for lambda_k for every nonzero target a_k, and in Example 2 to show a nonconstant entire function omits at most one value.
standard math Hurwitz's theorem: a locally uniform limit of locally univalent holomorphic functions is either constant or locally univalent.
Used in the proof of Theorem 3 to deduce that the limiting functions F_k are nonconstant and locally univalent once F0 is shown nonconstant.
standard math Iversen's theorem: every transcendental entire function has an asymptotic path to infinity along which it tends to infinity.
Implicit in Section 3 when asserting that 0 is an asymptotic value of each nonconstant nonvanishing entire function F_n, so that E_{(0,n)}(0) belongs to AV(F0) through equation (1).
domain assumption Standard classification of Fatou components for entire functions with finite singular set.
Example 3 concludes the Fatou set is empty from a repelling fixed point and a finite postsingular set. This requires additional standard facts, not stated in the paper.

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Cite this review

Pith. "Pith review of Entire functions with prescribed singular values." pith.science (2026). https://pith.science/paper/DMVMREZ4

@misc{pith2026190806026,
  author       = {Pith},
  title        = {Pith review of: Entire functions with prescribed singular values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMVMREZ4}},
  note         = {Machine review of arXiv:1908.06026}
}

abstract

We introduce a new class of entire functions $\mathcal{E}$ which consists of all $F_0\in\mathcal{O}(\mathbb{C})$ for which there exists a sequence $(F_n)\in \mathcal{O}(\mathbb{C})$ and a sequence $(\lambda_n)\in\mathbb{C}$ satisfying $F_n(z)=\lambda_{n+1}e^{F_{n+1}(z)}$ for all $n\geq 0$. This new class is closed under the composition and its is dense in the space of all non-vanishing entire functions. We prove that every closed set $V\subset \mathbb{C}$ containing the origin and at least one more point is the set of singular values of some locally univalent function in $\mathcal{E}$, hence this new class has non-trivial intersection with both the Speiser class and the Eremenko-Lyubich class of entire functions. As a consequence we provide a new proof of an old result by Heins which states that every closed set $V\subset\mathbb{C}$ is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally we show that the class $\mathcal{E}$ contains functions with an empty Fatou set and also functions whose Fatou set is non-empty.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Baker, P

I.N. Baker, P. Rippon: Iteration of exponential functions , Ann. Acad. Sci. Fenn., Ser. A I Math, (9) (1984), 49-77

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[2]

Bishop: Constructing entire functions by quasiconformal folding Acta Math., 214 (2015), 160

C.J. Bishop: Constructing entire functions by quasiconformal folding Acta Math., 214 (2015), 160

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Bergweiler, M

W. Bergweiler, M. Haruta, H. Kriete, H. Meier, N. Terglan e: On the limit functions of iterates in wandering domains . Ann. Acad. Sci. Fenn. Ser. A I Math.18(2),369375 (1993)

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A. E. Eremenko and M. Yu. Lyubich: Dynamical properties of some classes of entire functions , Ann. Inst. Fourier (Grenoble), 42 (1992), no. 4, 989-1020

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Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzw ert st (German), Math

W. Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzw ert st (German), Math. Ann.79 (12), 201-208 (1918)

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[6]

Heins: The set of asymptotic values of an entire function , Tolfte Skandinaviska Matematik- erkongressen (Lund, Sweden, 1953), Proceedings of the Scan dinavian Math

M. Heins: The set of asymptotic values of an entire function , Tolfte Skandinaviska Matematik- erkongressen (Lund, Sweden, 1953), Proceedings of the Scan dinavian Math. Congress, Lund, 1954, pp. 56-60

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Heins: Asymptotic spots of entire and meromorphic functions , Ann

M. Heins: Asymptotic spots of entire and meromorphic functions , Ann. Math. (2) 66, 430-439 (1957)

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[9]

Poincar´ e: Sur une classe ´ etendue de transcendantes uniformes

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H. Poincar´ e: Sur une classe nouvelle de transcendantes uniformes. J. Math. Pures Appl. IV. Ser. 6 (1890) 316365
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Schleicher, Dynamics of Entire Functions Holomorphic Dynamical Systems: Cetraro, Italy, July 7-12, 2008”, Springer Berlin Heidelberg, 295–3 39 (2010) L

D. Schleicher, Dynamics of Entire Functions Holomorphic Dynamical Systems: Cetraro, Italy, July 7-12, 2008”, Springer Berlin Heidelberg, 295–3 39 (2010) L. Boc Thaler: F aculty of Education, University of Ljubljana , SI–1000 Ljubljana, Slovenia. Institute of Mathematics, Physi...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Baker, P

I.N. Baker, P. Rippon: Iteration of exponential functions , Ann. Acad. Sci. Fenn., Ser. A I Math, (9) (1984), 49-77

work page 1984

[2] [2]

Bishop: Constructing entire functions by quasiconformal folding Acta Math., 214 (2015), 160

C.J. Bishop: Constructing entire functions by quasiconformal folding Acta Math., 214 (2015), 160

work page 2015

[3] [3]

Bergweiler, M

W. Bergweiler, M. Haruta, H. Kriete, H. Meier, N. Terglan e: On the limit functions of iterates in wandering domains . Ann. Acad. Sci. Fenn. Ser. A I Math.18(2),369375 (1993)

work page 1993

[4] [4]

A. E. Eremenko and M. Yu. Lyubich: Dynamical properties of some classes of entire functions , Ann. Inst. Fourier (Grenoble), 42 (1992), no. 4, 989-1020

work page 1992

[5] [5]

Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzw ert st (German), Math

W. Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzw ert st (German), Math. Ann.79 (12), 201-208 (1918)

work page 1918

[6] [6]

Heins: The set of asymptotic values of an entire function , Tolfte Skandinaviska Matematik- erkongressen (Lund, Sweden, 1953), Proceedings of the Scan dinavian Math

M. Heins: The set of asymptotic values of an entire function , Tolfte Skandinaviska Matematik- erkongressen (Lund, Sweden, 1953), Proceedings of the Scan dinavian Math. Congress, Lund, 1954, pp. 56-60

work page 1953

[7] [7]

Heins: Asymptotic spots of entire and meromorphic functions , Ann

M. Heins: Asymptotic spots of entire and meromorphic functions , Ann. Math. (2) 66, 430-439 (1957)

work page 1957

[8] [8]

Milnor: Dynamics in one complex variable , 3rd ed

J. Milnor: Dynamics in one complex variable , 3rd ed. Annals of Mathematics Studies 160, Princeton University Press, Princeton, NJ (2006)

work page 2006