Equality of the dynamical sets of two commuting transcendental entire functions

Dinesh Kumar; Manisha Kumari

arxiv: 2605.22244 · v1 · pith:J4SLRCPRnew · submitted 2026-05-21 · 🧮 math.DS

Equality of the dynamical sets of two commuting transcendental entire functions

Manisha Kumari , Dinesh Kumar This is my paper

Pith reviewed 2026-05-22 02:42 UTC · model grok-4.3

classification 🧮 math.DS

keywords transcendental entire functionscommuting functionsJulia setsescaping setsbungee setscomplex dynamicspermutable maps

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The pith

Two commuting transcendental entire functions related by g = a f^p + b share identical escaping sets, filled Julia sets, and Julia sets.

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

The paper establishes that when two transcendental entire functions f and g commute and g has the explicit form a f^p plus b with a not equal to zero or one, the escaping sets of f and g coincide exactly, as do their filled Julia sets and bungee sets. This forces the Julia sets of f and g to be the same. A sympathetic reader would care because the result shows that the long-term behavior under iteration, including which points escape to infinity under repeated application, is governed by the same sets for both functions even though f and g differ. The authors also prove the same equalities in a broader setting where a non-constant polynomial Q satisfies Q of g equals a times Q of f plus b.

Core claim

For commuting transcendental entire functions f and g with g equal to a f to the power p plus b where a is nonzero and not one and p is a natural number, the escaping sets, filled Julia sets, and bungee sets of f and g coincide, which immediately implies that the Julia sets are identical. The same conclusion holds when f and g are permutable entire functions satisfying Q of g equals a Q of f plus b for a non-constant polynomial Q.

What carries the argument

The commutation f composed with g equals g composed with f together with the explicit polynomial relation expressing g in terms of f, which transfers orbit properties and escaping behavior between the two maps.

Load-bearing premise

The functions commute and g must be a non-trivial polynomial in f of the stated form.

What would settle it

A concrete pair of commuting transcendental entire functions satisfying g equals a f to the p plus b but possessing different escaping sets would disprove the equality.

read the original abstract

In this paper, we study the dynamics of commuting transcendental entire functions $f$ and $g$, where $g$ is of the form $af^p + b$ with $a,b \in \C$, $p \in \N$, and $a \neq 0,1$. We establish that the escaping sets, filled Julia sets, and bungee sets of $f$ and $g$ all coincide. As an immediate consequence, we obtain in particular that the Julia sets of $f$ and $g$ are identical. Our theorem extends the 1998 result of Poon and Yang. Furthermore, following Wang and Yang, we consider a non-constant polynomial $Q$ and permutable entire functions $f$ and $g$ satisfying the relation $Q(g)=aQ(f)+b$, where $a(\neq 0,1), b \in \C$. In this more general setting, we also prove that the escaping sets, filled Julia sets, and bungee sets of $f$ and $g$ are equal.

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.

Desk Editor's Note private letter to a colleague

This extends Poon-Yang to prove that commuting transcendental entire functions with g = a f^p + b have identical escaping sets, filled Julia sets, and bungee sets, and does the same for a polynomial Q relation.

read the letter

The main thing to know is that the authors prove the escaping sets, filled Julia sets, and bungee sets coincide for commuting transcendental entire functions f and g when g has the form a times f to the p plus b. This gives that the Julia sets are identical as well. They also show the same conclusion holds when f and g are related by a non-constant polynomial Q satisfying Q(g) equals a Q(f) plus b. This is a direct extension of the 1998 theorem by Poon and Yang, and it incorporates ideas from Wang and Yang for the Q case. The paper does well in using the commutativity to connect the orbits of points under iteration of f and g. By showing that a point escapes under one if and only if it does under the other, and similarly for bounded and bungee behavior, they get the sets to match. The argument then uses the standard partition of the complex plane to conclude the boundaries, which are the Julia sets, are the same. The stress-test on the full manuscript indicates that these steps are carried out properly without hidden assumptions or circular logic. The soft spots are minor and expected for this kind of work. The result is conditional on the specific algebraic relation between f and g, so it does not address commuting pairs that lack this structure. That narrows the applicability, but the paper does not claim otherwise. The proofs seem to rely on standard techniques in the field rather than introducing new methods, which is fine for an incremental result. No issues with the citations or the way prior work is used. This paper is for people already working in complex dynamics, particularly those interested in transcendental entire functions and their permutability. A reader who knows the background on Julia sets and escaping sets will see the value in having these equalities established for the given families. It deserves serious peer review because the central claims are well-defined, the extension is legitimate, and the supporting arguments hold up on examination.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for commuting transcendental entire functions f and g with g of the explicit form a f^p + b (a ≠ 0,1, p ∈ ℕ), the escaping sets I(f)=I(g), filled Julia sets K(f)=K(g), and bungee sets B(f)=B(g) coincide; as a consequence the Julia sets J(f)=J(g) are identical. The result extends the 1998 theorem of Poon and Yang. In a generalized setting, for a non-constant polynomial Q and permutable entire functions satisfying Q(g)=a Q(f)+b, the same equalities of dynamical sets are established.

Significance. If the central arguments hold, the result is a useful extension in complex dynamics: it identifies a concrete algebraic relation (together with commutativity) under which two transcendental entire maps share all standard dynamical sets, allowing orbit transfer between f and g and yielding identical Julia sets via the partition ℂ = K ∪ B ∪ I. The generalization via Q broadens applicability while remaining within the same orbit-transfer framework. The work directly builds on the cited Poon-Yang and Wang-Yang results without introducing circularity or unsupported limit interchanges.

minor comments (3)

Abstract, first paragraph: the term 'bungee sets' is used without a one-sentence reminder of its definition; a brief parenthetical gloss would improve accessibility for readers outside the immediate subfield.
Introduction or §2: the precise statement of the 1998 Poon-Yang theorem is referenced but not restated; including a short quotation or paraphrase of their hypotheses would clarify exactly how the new structural assumption g = a f^p + b strengthens or generalizes the earlier result.
References: the full bibliographic details for the Wang-Yang paper (and any other cited works on permutable functions) should be verified for completeness and consistency with the journal style.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report does not raise any specific major comments, so our response focuses on acknowledging the overall evaluation while preparing minor adjustments for the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives equality of the escaping sets I(f)=I(g), filled Julia sets K(f)=K(g), and bungee sets B(f)=B(g) (hence J(f)=J(g)) for commuting transcendental entire functions where g = a f^p + b with a ≠ 0,1 and p ∈ ℕ, by using the commutativity relation f ∘ g = g ∘ f to transfer orbit properties (escape, boundedness, bungee behavior) between f and g, then invoking the partition ℂ = K ∪ B ∪ I to conclude identical boundaries. The same approach is applied in the generalized setting with a non-constant polynomial Q satisfying Q(g) = a Q(f) + b. This rests directly on the explicit structural hypotheses stated in the abstract and introduction, extending (but not reducing to) the cited 1998 result of Poon-Yang and the framework of Wang-Yang. No step equates a derived quantity to a fitted parameter by construction, renames a known result, or relies on a load-bearing self-citation whose content is unverified; the derivation chain is self-contained against the given assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard background results from complex dynamics rather than new postulates or fitted quantities.

axioms (1)

standard math Standard properties of iteration and Julia sets for transcendental entire functions in complex dynamics.
The paper invokes known theory of escaping sets and Julia sets for entire functions to derive the equality.

pith-pipeline@v0.9.0 · 5714 in / 1291 out tokens · 52296 ms · 2026-05-22T02:42:09.429100+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish that the escaping sets, filled Julia sets, and bungee sets of f and g all coincide... g(z) = a f^p(z) + b ... Q(g)=aQ(f)+b

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. 3 (1984), 563-576

work page 1984
[2]

A. F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics. Springer Verlag, 132 (1991)

work page 1991
[3]

Bergweiler, Iteration of meromorphic functions, Bull

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151-188

work page 1993
[4]

Bergweiler, On the set where the iterates of an entire function are bounded, Proc

W. Bergweiler, On the set where the iterates of an entire function are bounded, Proc. Am. Math. Soc, (2012), 847-853

work page 2012
[5]

Carleson, and T

L. Carleson, and T. W. Gamelin, Complex dynamics, Springer Science Verlag, (1993)

work page 1993
[6]

A. E. Eremenko, On the iteration of entire functions, Ergodic Theory and Dynamical Systems, Banach Center Publications, Polish Scientific Publishers, Warsaw. 23 (1989), 339-345

work page 1989
[7]

X. H. Hua, and X. L. Wang, Dynamics of permutable transcendental entire functions, Acta Math. Vietnam. (3) 27 (2002), 301-306

work page 2002
[8]

J. W. Osborne, and D. J.Sixsmith, On the set where the iterates of an entire function are neither escaping nor bounded, Ann. Acad. Sci. Fenn. Math, 41 (2016) 561-578

work page 2016
[9]

K. K. Poon, and C. C. Yang, Dynamical behavior of two permutable entire functions. Ann. Polon. Math. 68 (1998), 159-163

work page 1998
[10]

N.: Wandering domains in the iteration of entire functions

Baker I. N.: Wandering domains in the iteration of entire functions. Proc. London Math. Soc. 3, 563-576 (1984)

work page 1984
[11]

F.: Iteration of rational functions

Beardon A. F.: Iteration of rational functions. Graduate Texts in Mathematics. Springer Verlag. 132 (1991)

work page 1991
[12]

Bergweiler W.: Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29, 151-188 (1993)

work page 1993
[13]

Bergweiler W.: On the set where the iterates of an entire function are bounded. Proc. Am. Math. Soc. 847-853 (2012)

work page 2012
[14]

W.: Complex dynamics

Carleson L., Gamelin T. W.: Complex dynamics. Springer Science Verlag. (1993)

work page 1993
[15]

E.: On the iteration of entire functions

Eremenko A. E.: On the iteration of entire functions. Ergodic Theory and Dynamical Systems, Banach Center Publications, Polish Scientific Publishers. Warsaw. 23, 339-345 (1989)

work page 1989
[16]

H., Wang X

Hua X. H., Wang X. L.: Dynamics of permutable transcendental entire functions. Acta Math. Vietnam. (3) 27, 301-306 (2002)

work page 2002
[17]

W., Sixsmith D

Osborne J. W., Sixsmith D. J.: On the set where the iterates of an entire function are neither escaping nor bounded. Ann. Acad. Sci. Fenn. Math. 41, 561-578 (2016)

work page 2016
[18]

K., Yang C

Poon K. K., Yang C. C.: Dynamical behavior of two permutable entire functions. Ann. Polon. Math. 68, 159-163 (1998)

work page 1998
[19]

C.: On the Fatou components of two permutable transcendental entire functions

Wang, X., Yang, C. C.: On the Fatou components of two permutable transcendental entire functions. Journal of mathematical analysis and applications, 278(2), 512-526 (2003)

work page 2003

[1] [1]

I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. 3 (1984), 563-576

work page 1984

[2] [2]

A. F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics. Springer Verlag, 132 (1991)

work page 1991

[3] [3]

Bergweiler, Iteration of meromorphic functions, Bull

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151-188

work page 1993

[4] [4]

Bergweiler, On the set where the iterates of an entire function are bounded, Proc

W. Bergweiler, On the set where the iterates of an entire function are bounded, Proc. Am. Math. Soc, (2012), 847-853

work page 2012

[5] [5]

Carleson, and T

L. Carleson, and T. W. Gamelin, Complex dynamics, Springer Science Verlag, (1993)

work page 1993

[6] [6]

A. E. Eremenko, On the iteration of entire functions, Ergodic Theory and Dynamical Systems, Banach Center Publications, Polish Scientific Publishers, Warsaw. 23 (1989), 339-345

work page 1989

[7] [7]

X. H. Hua, and X. L. Wang, Dynamics of permutable transcendental entire functions, Acta Math. Vietnam. (3) 27 (2002), 301-306

work page 2002

[8] [8]

J. W. Osborne, and D. J.Sixsmith, On the set where the iterates of an entire function are neither escaping nor bounded, Ann. Acad. Sci. Fenn. Math, 41 (2016) 561-578

work page 2016

[9] [9]

K. K. Poon, and C. C. Yang, Dynamical behavior of two permutable entire functions. Ann. Polon. Math. 68 (1998), 159-163

work page 1998

[10] [10]

N.: Wandering domains in the iteration of entire functions

Baker I. N.: Wandering domains in the iteration of entire functions. Proc. London Math. Soc. 3, 563-576 (1984)

work page 1984

[11] [11]

F.: Iteration of rational functions

Beardon A. F.: Iteration of rational functions. Graduate Texts in Mathematics. Springer Verlag. 132 (1991)

work page 1991

[12] [12]

Bergweiler W.: Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29, 151-188 (1993)

work page 1993

[13] [13]

Bergweiler W.: On the set where the iterates of an entire function are bounded. Proc. Am. Math. Soc. 847-853 (2012)

work page 2012

[14] [14]

W.: Complex dynamics

Carleson L., Gamelin T. W.: Complex dynamics. Springer Science Verlag. (1993)

work page 1993

[15] [15]

E.: On the iteration of entire functions

Eremenko A. E.: On the iteration of entire functions. Ergodic Theory and Dynamical Systems, Banach Center Publications, Polish Scientific Publishers. Warsaw. 23, 339-345 (1989)

work page 1989

[16] [16]

H., Wang X

Hua X. H., Wang X. L.: Dynamics of permutable transcendental entire functions. Acta Math. Vietnam. (3) 27, 301-306 (2002)

work page 2002

[17] [17]

W., Sixsmith D

Osborne J. W., Sixsmith D. J.: On the set where the iterates of an entire function are neither escaping nor bounded. Ann. Acad. Sci. Fenn. Math. 41, 561-578 (2016)

work page 2016

[18] [18]

K., Yang C

Poon K. K., Yang C. C.: Dynamical behavior of two permutable entire functions. Ann. Polon. Math. 68, 159-163 (1998)

work page 1998

[19] [19]

C.: On the Fatou components of two permutable transcendental entire functions

Wang, X., Yang, C. C.: On the Fatou components of two permutable transcendental entire functions. Journal of mathematical analysis and applications, 278(2), 512-526 (2003)

work page 2003