On a question of Astorg and Boc Thaler
Pith reviewed 2026-05-17 04:56 UTC · model grok-4.3
The pith
When α is algebraic, θ must lie in (1/P(1))ℤ for the sequence σ_k to converge.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
When α is an algebraic number, denoting by P(x) ∈ ℤ[x] the minimal polynomial of α, we prove that θ ∈ (1/P(1))ℤ is necessary and sufficient for the existence of (n_k) such that (σ_k) converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on ℂ² with wandering domains of rank one.
What carries the argument
The minimal polynomial P(x) ∈ ℤ[x] of the algebraic number α, which fixes the precise lattice (1/P(1))ℤ that θ must occupy to allow a convergent sequence σ_k.
If this is right
- Whenever θ lies in the lattice (1/P(1))ℤ, the skew-product admits wandering domains of rank one.
- The necessity of the lattice condition holds for every algebraic α > 1.
- New explicit skew-products on ℂ² with rank-one wandering domains are obtained by choosing algebraic α and β such that θ satisfies the lattice condition.
- The Pisot case treated earlier is recovered when P(1) = 1.
Where Pith is reading between the lines
- The lattice condition on θ may continue to be sufficient for certain transcendental α, although necessity is not addressed.
- Varying α through algebraic numbers could trace the boundary between maps that possess wandering domains and those that do not.
- Minimal-polynomial techniques may apply to other holomorphic maps where integer sequences must accumulate in a controlled way.
Load-bearing premise
The assumption that α is algebraic, so that its minimal polynomial exists and governs the arithmetic of possible convergent sequences.
What would settle it
An explicit algebraic α together with a θ not in (1/P(1))ℤ for which some increasing sequence n_k still makes σ_k converge would disprove the necessity direction.
read the original abstract
Astorg and Boc Thaler studied the dynamics of certain skew-product tangent to the identity on $\mathbb{C}^2$, with two real parameters $\alpha>1$ and $\beta$ derived from its coefficients. They proved that if there exists an increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}:=(n_{k+1}-\alpha n_k-\beta\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $\alpha>1$ with the Pisot property, the condition that $\theta:=\frac{\beta\ln\alpha}{\alpha-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $\alpha$ is an algebraic number, we answer the question of Astorg and Boc Thaler in the affirmative. Furthermore, denoting by $P(x)\in\mathbb{Z}[x]$ the minimal polynomial of~$\alpha$, we prove that $\theta\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on $\mathbb{C}^2$ with wandering domains of rank one.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper resolves a question of Astorg and Boc Thaler on skew-product maps tangent to the identity on ℂ². For algebraic α > 1 with minimal polynomial P(x) ∈ ℤ[x], the authors prove that θ := β ln α / (α − 1) lies in (1/P(1))ℤ if and only if there exists an increasing sequence of positive integers (n_k) such that the sequence σ_k := n_{k+1} − α n_k − β ln n_k converges. Combined with the earlier sufficiency result of Astorg and Boc Thaler, this yields explicit new examples of such skew-products possessing wandering domains of rank one.
Significance. If the result holds, it supplies a sharp arithmetic criterion characterizing the parameters for which wandering domains of rank one exist in this family, thereby furnishing concrete new examples and completing the necessity direction under the algebraicity hypothesis. The proof strategy—treating the recurrence as an inhomogeneous linear relation annihilated by the minimal polynomial and extracting the consistency condition on the linear-in-k term—is a direct application of standard algebraic-number techniques to a dynamical-systems question.
major comments (1)
- [§3] §3, proof of necessity: the argument derives the condition θ ∈ (1/P(1))ℤ by applying the minimal polynomial to the approximate recurrence and isolating the coefficient of the linear term generated by ln n_k ∼ k ln α. It would be helpful to make explicit the error-control step that shows the remainder term remains bounded when the arithmetic condition holds (or diverges otherwise), as this step appears load-bearing for the necessity claim.
minor comments (3)
- [Introduction / Theorem statement] The statement of the main theorem (presumably Theorem 1.1 or 2.1) should explicitly record that α > 1 is algebraic and that P is its minimal polynomial over ℤ; the current abstract phrasing is slightly ambiguous on whether the result extends verbatim to non-monic polynomials.
- [§4] In the sufficiency construction, the recursive definition of n_{k+1} should include a brief verification that the sequence remains strictly increasing for all sufficiently large k; this is presumably straightforward but is not flagged in the text.
- [Introduction] A short remark comparing the new arithmetic condition θ ∈ (1/P(1))ℤ with the earlier rationality condition of Astorg–Boc Thaler (for Pisot α) would clarify the improvement for the reader.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. The single major comment is addressed below; we will revise the paper to incorporate the suggested clarification.
read point-by-point responses
-
Referee: [§3] §3, proof of necessity: the argument derives the condition θ ∈ (1/P(1))ℤ by applying the minimal polynomial to the approximate recurrence and isolating the coefficient of the linear term generated by ln n_k ∼ k ln α. It would be helpful to make explicit the error-control step that shows the remainder term remains bounded when the arithmetic condition holds (or diverges otherwise), as this step appears load-bearing for the necessity claim.
Authors: We agree that an explicit error-control argument would improve the clarity and transparency of the necessity proof in §3. While the existing derivation applies the minimal polynomial to the recurrence and isolates the linear coefficient arising from the asymptotic ln n_k ∼ k ln α, the bounds on the resulting remainder term are currently only implicit. In the revised version we will insert a short paragraph immediately after the isolation of the linear term, in which we estimate the remainder E_k explicitly using the fact that n_k grows exponentially (n_k ∼ C α^k) and the logarithmic perturbation is o(k). We will show that |E_k| stays bounded precisely when the coefficient of the linear term vanishes, i.e., when θ ∈ (1/P(1))ℤ, and diverges otherwise. This makes the necessity direction fully rigorous without changing any of the main statements or the overall proof strategy. revision: yes
Circularity Check
No significant circularity identified
full rationale
The paper's central derivation applies the minimal polynomial P of the algebraic number α directly to the approximate recurrence relation n_{k+1} ≈ α n_k + β ln n_k, using the annihilator property of P to extract a consistency condition on the linear-in-k term generated by ln n_k ∼ k ln α. This forces the arithmetic constraint θ ∈ (1/P(1))ℤ as a necessary condition, with sufficiency shown by explicit recursive construction inside the lattice generated by P. The argument relies on standard properties of algebraic numbers and linear recurrences rather than any fitted parameter renamed as prediction, self-citation chain, or imported uniqueness theorem. The cited work of Astorg and Boc Thaler is external and provides only the dynamical interpretation, not the arithmetic necessity proof itself. The derivation is therefore self-contained against the stated hypotheses.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Properties of minimal polynomials of algebraic numbers over the integers
- domain assumption Existence and basic arithmetic of sequences (n_k) satisfying the convergence of σ_k when the rationality condition holds
Forward citations
Cited by 1 Pith paper
-
Distribution modulo one of linear recurrent sequences
Criteria are established for finite limit sets of fractional parts of linear recurrent sequences, with lower bounds on maximal distances between limit values.
Reference graph
Works this paper leans on
-
[1]
Astorg , M.; Buff , X.; Dujardin , R.; Peters , H.; Raissy , J.: A two-dimensional polynomial mapping with a wandering Fatou component. Ann. of Math. (2) 184 (2016), no. 1, 263–313
work page 2016
-
[2]
B.: Dynamics of skew-products tangent to the identity
Astorg M.; Thaler L. B.: Dynamics of skew-products tangent to the identity. J. Eur. Math. Soc., (2024), published online first
work page 2024
-
[3]
Cambridge Tracts in Math., 193 Cambridge University Press, Cambridge, (2012)
Bugeaud , Y.: Distribution modulo one and Diophantine approximation. Cambridge Tracts in Math., 193 Cambridge University Press, Cambridge, (2012). xvi+300 pp. ISBN:978-0-521-11169-0
work page 2012
-
[4]
France , M.: Les suites à spectre vide et la répartition modulo 1. J. of number theory, t. 5, 1973, p. 1-15
work page 1973
-
[5]
Pisot , C.: Répartition modulo 1 des puissances successives des nombres r\'eels. Comment. Math. Helv. 19, (1946), 153–160
work page 1946
-
[6]
Sullivan , D.: Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains. Ann. of Math. (2) 122 (1985), no. 3, 401–418
work page 1985
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.