Singularities Admitting Contracting Automorphisms

K\'emo Morvan (UPCit\'e)

REVIEW 4 major objections 6 minor 14 references

Singularities Admitting Contracting Automorphisms

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Any complex singularity with a contracting automorphism is quasi-homogeneous.

desk verdict A genuine higher-dimensional generalization of Favre-Ruggiero; the main proof is sound, with two fixable gaps in exposition. read the letter →

arxiv 2412.11583 v1 pith:CPS7PRVZ submitted 2024-12-16 math.CV

Kémo Morvan (UPCité) This is my paper · ORCID

classification math.CV MSC 32S0537F9932B1014B0532A10

keywords contractingautomorphismquasi-homogeneoussingularityweightedhomogeneouspolynomialholomorphicdynamicsC*-actionPoincaré-Dulacnormalforminvariantidealcomplexanalytic

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 proves that a complex singularity that admits a contracting automorphism—a holomorphic self-map fixing the origin, invertible, and shrinking a whole neighborhood into itself so that every orbit converges to the origin—must be quasi-homogeneous, meaning it can be embedded so that its defining ideal is generated by weighted homogeneous polynomials. This concludes, in all dimensions and without normality assumptions, a characterization that was previously known only for normal surface singularities and for two-dimensional Stein spaces. The result connects the dynamical existence of a contracting symmetry to the existence of a $C^*$-action on the singularity. The proof avoids resolution of singularities: it embeds the germ minimally, extends the automorphism to a contracting automorphism of the ambient space, and then shows that any invariant ideal under such an ambient contraction is generated by weighted homogeneous polynomials.

What carries the argument

The argument is carried by two devices. The first is the theory of good embeddings: when a singularity germ is embedded in $\mathbb{C}^d$ with $d$ equal to its embedding dimension, every extension of an automorphism of the germ is an ambient automorphism, and every extension of a contracting automorphism is an ambient contracting automorphism (Theorem 3.3). The proof of the contracting part uses the stable manifold theorem together with Cauchy estimates on holomorphic arcs that leave the stable manifold, forcing all eigenvalues of the ambient derivative to have modulus $<1$. The second device is the $\lambda$-gradation built from the spectrum of a contracting ambient automorphism in Poincaré-Dulac normal form: the $\lambda$-order on $\mathbb{N}^d$ organizes monomials into finite-dimensional spaces $H^\gamma$ of $\lambda$-homogeneous polynomials, and lemmas 4.15 and 4.16 show that these spaces are permuted and stably acted on by the automorphism. The main algebraic result (Theorem 4.1) uses a double inclusion and a triangular linear algebra lemma to prove that any invariant ideal is generated by such $\lambda$-homogeneous polynomials.

What would settle it

An explicit counterexample would be a non-quasi-homogeneous singularity, for instance a non-reduced germ or a union of branches with incompatible weights, that still admits a contracting automorphism; the first place to look is where the proof's curve-selection step fails, namely a reducible germ in which no holomorphic arc through a point off the shrinking direction lies entirely in the singularity.

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Extended reading notes

Core claim

The central claim is Theorem A: if $(X,0)$ is a complex analytic singularity and $f:(X,0)\to(X,0)$ is a contracting automorphism, then $(X,0)$ is quasi-homogeneous. Since every quasi-homogeneous singularity carries contracting automorphisms through its $C^*$-action, the theorem gives a full characterization of when contracting automorphisms exist. The proof splits into two statements that are of independent interest. Theorem 3.3 shows that in an embedding of minimal dimension, every extension of a contracting automorphism of the singularity is a contracting automorphism of the ambient space. Theorem 4.1 shows that any ideal of holomorphic functions invariant under a contracting diffeomorphism of $\mathbb{C}^d$ in Poincaré-Dulac normal form is generated by weighted homogeneous polynomials, with an invariant filtration of the ideal preserved. Applied to the defining ideal of the singularity, these two statements yield Theorem A.

Load-bearing premise

The load-bearing premise is that for every point of the singularity off the shrinking direction one can draw a holomorphic arc inside the singularity from the origin through that point, and the automorphism shrinks the whole circle trace of that arc uniformly; this curve selection is assumed rather than proven for arbitrary (possibly reducible) singularities.

Editorial extensions

If this is right

Every complex singularity that admits a contracting automorphism is quasi-homogeneous, so it also carries a $C^*$-action and weighted-homogeneous defining equations.
The dimension-two results of Favre and Ruggiero and of Camacho, Movasati, and Scárdua become special cases of a single theorem valid for arbitrary germs.
Theorem 4.1 provides a normal form for invariant ideals under contracting diffeomorphisms: any such ideal has a filtered generating set of $\lambda$-homogeneous polynomials.
Theorem 3.3 shows that the embedding dimension is the correct invariant for extension problems: in minimal embeddings, contractions of the germ extend to contractions of the ambient space.

Reading between the lines

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

A natural testable extension is to replace 'automorphism' by 'endomorphism': the Poincaré-Dulac and invariant-ideal arguments appear to need only a contracting self-map, so the same conclusion may hold without invertibility.
The $\lambda$-gradation machinery could be applied to ideals invariant under a semigroup of commuting contractions, which would give a new way to prove quasi-homogeneity for singularities with a $\mathbb{C}^*$-action without resolving them.
Combining Theorem 4.1 with Saito's criterion would make every singularity with a contracting automorphism have equal Milnor and Tjurina numbers, and conversely a dynamical proof of quasi-homogeneity from that equality.
Since the proof avoids resolution, it may carry over to formal or real-analytic germs where resolution is unavailable, provided the stable manifold theorem and Poincaré-Dulac normal form have analogues.

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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

Kémo Morvan has a real theorem here: it removes the dimension restriction from the Favre-Ruggiero result and shows that a contracting automorphism forces quasi-homogeneity in arbitrary dimension. That is a natural open question and this is a genuine advance. The proof strategy is sensible—extend to a minimal embedding, use Poincaré-Dulac normal form, then analyze invariant ideals via a λ-gradation. Section 4 is the real novelty: the gradation by eigenvalue products and the double-inclusion argument for ideals of arbitrary codimension are thoughtful and mostly convincing.

The two soft spots the reader flagged are real but smaller than they look. In Theorem 3.3, the curve φ is not explicitly required to lie inside X until the moment it is used. That is a one-line fix: take the attracting neighborhood U from Definition 2.1, pick p in U ∩ ι(X,0), and choose a curve through p inside U. The contraction property then gives the swallowed loop exactly as claimed. The paper does not say this, and a referee should ask for it, but it is not a load-bearing error.

The second worry, about passing from λ-homogeneous generators to standard weighted homogeneous generators, also has a clean patch. The equivalence classes are the cosets of the resonance lattice L = {α−β : λ^α = λ^β}. Since |λ_i|<1, L contains no nonzero vector in the nonnegative orthant, so L^⊥ contains a vector with all positive integer entries. That vector gives the weight system. The paper does not spell this out; it just asserts the conclusion. Same diagnosis: fixable, needs a few lines.

I do not see circularity or hidden fitting. The citations to Favre-Ruggiero, Orlik-Wagreich, and the normal form literature are appropriate. The main theorem is the genuine route to the result.

This paper deserves a serious referee. It is a strong contribution to singularity theory and holomorphic dynamics. The referee will need to ask for the two clarifications above, and maybe for a careful rewriting of Proposition 2.3, but the core is in good shape. I would send it out.

Referee Report

4 major / 6 minor

Summary. The paper claims Theorem A: if a complex analytic singularity (X,0) admits an automorphism f that is contracting in the sense of Definition 2.1, then (X,0) is quasi-homogeneous. The proof has two main steps. First, Theorem 3.3 asserts that in an embedding of minimal dimension (a "good" embedding), any extension of f to the ambient (C^d,0) is automatically a contracting automorphism of the ambient space. This is proved by contradiction using the stable manifold theorem and a holomorphic curve through a point outside the stable manifold. Second, Theorem 4.1 asserts that any ideal invariant under a contracting automorphism of (C^d,0) is generated by weighted homogeneous polynomials. This is proved by introducing a λ-order and λ-gradation adapted to the linear part of the automorphism, and then running a double induction on the order and the number of generators. The paper concludes that the ideal of (X,0) is weighted homogeneous, hence the singularity is quasi-homogeneous.

Significance. If correct, this is a strong result: it extends the two-dimensional results of Favre–Ruggiero and Camacho–Movasati–Scárdua to arbitrary dimensions and to arbitrary (possibly reducible) singularities, giving a complete dynamical characterization of quasi-homogeneity. The λ-gradation machinery in Section 4 is a potentially useful combinatorial tool for invariant ideals under contracting automorphisms. The paper relies on standard external results (stable manifold theorem, Poincaré–Dulac normal forms), has no free parameters, and does not appear to be circular. However, several load-bearing steps in the proofs are not justified as written, so the significance is conditional on these gaps being repaired.

major comments (4)

[Theorem 3.3, §3.2] The proof chooses a holomorphic curve φ: (C,0)→(C^d,0) with φ(0)=0 and p∈Im(φ), but it does not show that Im(φ)⊂ι(X,0). The subsequent step "Let C=φ(S^1). As it is compact and F is contracting on X, we know that for all ε>0 there exists m such that F^m(C)⊂B(0,ε)" applies the contraction property of F on X to the loop C, which is only justified if C⊂ι(X,0) and C lies inside an attracting neighborhood U as in Definition 2.1. Since p is chosen outside the stable manifold, it is not automatic that a curve in X through 0 and p exists or that its trace on a circle around 0 is contained in the basin of attraction. Please provide a proof of the required curve selection inside X (e.g., via desingularization or normalization) and explain how p is chosen small enough so that C⊂U. Without this, the estimate leading to |λ_l|<1 does not follow.
[§2.1, Proposition 2.3] The proof that every contracting automorphism is uniformly contracting is defective. The sequence (x_n) is not shown to converge; only a subsequence converges, and the limit of a sequence in X\V need not lie outside V (it may lie on the boundary). The statement itself is true, and a correct proof can be obtained by noting that the compact sets F^n(K) are nested with intersection {0}, so they must eventually be contained in any neighborhood of 0. Since the uniform contraction property is later used to justify that the compact set C is swallowed by arbitrarily small balls, this proof should be rewritten.
[§4.2, Definition 4.7 and Remark 4.10(2)] The claim that the λ-order makes (Γ,⪰) order isomorphic to (N,≥) is false for generic eigenvalues. When the numbers |λ_i| are rationally independent (e.g., λ_1=2^{-√2}, λ_2=2^{-√3}), the set {|λ^{-α}|: α∈N^d} is dense in [1,∞), so the λ-order is not a well-order. Consequently, the induction over Γ in Proposition 4.22, which refers to "the preceding pair (j,δ)", is not justified; a transfinite induction would be needed, and the order type is not N. This is a load-bearing issue for the combinatorial proof of Theorem 4.1, since the invariant-ideal argument depends on this induction.
[Theorem 4.1 and §4.5–4.6] Theorem 4.1 concludes that I is generated by weighted homogeneous polynomials in the sense of Definition 2.6, which requires a single integer weight vector (n_1,...,n_d) with n_i≠0. The proof, however, establishes generation by λ-homogeneous polynomials with respect to the gradation H^γ. It is not shown that these polynomials are weighted homogeneous for a common integer weight vector. A necessary additional step is to prove that the lattice L={m∈Z^d: λ^m=1} has a rational orthogonal complement containing a vector with all coordinates nonzero; this follows from |λ_i|<1, but it is not mentioned. Please add this argument, as Definition 2.7 and Theorem A depend on it.

minor comments (6)

[§3.2, proof of Theorem 3.3] The Cauchy estimate has a numerical factor error: the final bound should be ε, not ε/(2π). This does not affect the conclusion that |λ_l|<1, but it should be corrected.
[§4.2, Definition 4.8] The text says "two n-tuples α, α′" but the ambient dimension is d; this should read "d-tuples".
[§4.4, Lemma 4.26] The auxiliary map used to define the partial order is denoted φ, which conflicts with the holomorphic curve φ in Theorem 3.3. Using a different letter would avoid confusion.
[Theorem 4.1 and §4.5] The proof of Proposition 4.22 uses ϕ^(i)(0)=0, which is only justified if I is an ideal of functions vanishing at 0, i.e., the ideal of a subvariety through 0. If Theorem 4.1 is intended for arbitrary invariant ideals, this restriction should be stated explicitly; otherwise, the theorem statement should say I⊂m_{C^d,0}.
[Throughout] The manuscript does not explicitly state that the singularity (X,0) is reduced. Several arguments (e.g., analytic continuation of the stable manifold intersection, use of open subsets of X) implicitly require reducedness. Please state the reducedness assumption or adjust the proofs to cover the non-reduced case.
[Section title] There is a typo in the heading of Section 4: "Inv ariant ideals" should be "Invariant ideals".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the theorem is a pure mathematical derivation from external standard results, with no fitted parameters and no conclusion assumed by construction.

full rationale

The paper proves a mathematical theorem and contains no empirical predictions, fitted parameters, or data-fitting steps. The central claim, Theorem A, is that a singularity admitting a contracting automorphism is quasi-homogeneous. The proof proceeds in two independent stages. First, Theorem 3.3 shows that a contracting automorphism of a singularity extends to a contracting automorphism of the ambient space with respect to a minimal embedding. This uses the stable manifold theorem and an elementary contradiction argument with Cauchy estimates; it does not invoke the quasi-homogeneity of (X,0). Second, Theorem 4.1 shows that any invariant ideal under a contracting diffeomorphism in Poincaré-Dulac normal form is generated by weighted homogeneous polynomials, which is exactly quasi-homogeneity. The construction of the λ-order, the λ-gradation, and the polynomials P(i) is internal to the proof and does not presuppose the conclusion. The cited external tools—Poincaré-Dulac normal forms, the stable manifold theorem, and standard facts on embedding dimension—are independent mathematical results, not self-citations and not equivalent to the target theorem. Prior works [4] and [6] are mentioned only as context and as results to be generalized; their conclusions are not used as premises. There is no self-citation chain, no fitted input renamed as a prediction, and no uniqueness theorem imported from the authors. The only possible concern is a proof gap in Theorem 3.3 regarding whether an arbitrary compact loop C=φ(S^1) is swallowed by the contracting dynamics, but even if that step were unproved, it would be a correctness issue, not circularity: it does not assume the conclusion or reduce the theorem to its inputs. Therefore the derivation is self-contained with respect to circularity, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper is a pure theorem in local holomorphic dynamics. It introduces no fitted parameters and no new physical or geometric entities. It relies on standard results such as Poincaré-Dulac normal forms, the stable manifold theorem, and Cauchy estimates. Two implicit geometric inputs are flagged: curve selection in X and the existence of integer weights for λ-homogeneous polynomials.

assumptions (7)

standard math Poincaré-Dulac theorem: any contracting automorphism of (C^d,0) is holomorphically conjugate to a map in Poincaré-Dulac normal form.
Invoked as Theorem 4.5 and used throughout Section 4 to restrict attention to normal forms.
standard math Stable manifold theorem for contracting holomorphic diffeomorphisms.
Used in Section 3.2 to obtain the invariant stable manifold and the splitting of eigenvalues.
standard math Contraction principle on (C^d,0): a diffeomorphism is contracting iff all eigenvalues of D0F have modulus less than 1.
Used in Sections 2.1 and 3.2 to relate contraction to eigenvalues; cited to [8].
domain assumption Curve selection: for a point p in a reduced complex analytic germ (X,0), there exists a holomorphic curve in X through 0 and p.
Implicitly used in the proof of Theorem 3.3 when applying contraction to φ(S^1); not stated or proved in the paper.
domain assumption Existence of a positive integer weight vector orthogonal to the resonance lattice L={ν∈Z^d : λ^ν=1}.
Needed to conclude that λ-homogeneous polynomials are weighted homogeneous in the sense of Definition 2.7; not proved or cited in the paper.
standard math Nakayama-style fact: a minimal set of generators of a proper ideal in a local ring has all syzygy coefficients in the maximal ideal.
Used in Lemma 4.20 to show the constant part A0 of the relation matrix is uniquely determined.
standard math Cauchy estimates for holomorphic functions on compact circles.
Used in Section 3.2 to bound the coefficient of the leading term in iterates of the curve φ.

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Cite this review

Pith. "Pith review of Singularities Admitting Contracting Automorphisms." pith.science (2026). https://pith.science/paper/CPS7PRVZ

@misc{pith2026241211583,
  author       = {Pith},
  title        = {Pith review of: Singularities Admitting Contracting Automorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPS7PRVZ}},
  note         = {Machine review of arXiv:2412.11583}
}

abstract

Given a complex analytic singularity $(X, 0)$, we show that if there exists an automorphism $F: (X, 0) \to (X, 0)$ that is contracting, then $(X, 0)$ is quasi-homogeneous.

Figures

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

An introduction to hyperbolic dynamical systems

Marco Abate. An introduction to hyperbolic dynamical systems. 2001

work page 2001

[2] [2]

Méthodes de changement d’échelles en analyse complexe.Ann

François Berteloot. Méthodes de changement d’échelles en analyse complexe.Ann. Fac. Sci. Toulouse Math. (6), 15(3):427–483, 2006

work page 2006

[3] [3]

Bruce-roberts numbers and quasihomogeneous functions on analytic varieties, 2022

Carles Bivià-Ausina, Konstantinos Kourliouros, and Maria Aparecida Soares Ruas. Bruce-roberts numbers and quasihomogeneous functions on analytic varieties, 2022

work page 2022

[4] [4]

Movasati, and Bruno Scardua

Cesar Camacho, H. Movasati, and Bruno Scardua. The moduli of quasi-homogeneous stein surface singular- ities. Journal of Geometric Analysis, 19:244–260, 04 2009

work page 2009

[5] [5]

H. Dulac. Solutions d’un système d’équations différentielles dans le voisinage de valeurs singulières.Bulletin de la Société Mathématique de France, 40:324–383, 1912

work page 1912

[6] [6]

Normal surface singularities admitting contracting automorphisms.Ann

Charles Favre and Matteo Ruggiero. Normal surface singularities admitting contracting automorphisms.Ann. Fac. Sci. Toulouse, Math. (6), 23(4):797–828, 2014

work page 2014

[7] [7]

Robert C. Gunning. Introduction to holomorphic functions of several variables. Vol. I. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory

work page 1990

[8] [8]

Morosawa, Y

S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda.Holomorphic dynamics, volume 66 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2000. Translated from the 1995 Japanese original and revised by the authors

work page 2000